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First published online March 28, 2008
Journal of Experimental Biology 211, 1221-1230 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.010652
Three-dimensional flow structures and evolution of the leading-edge vortices on a flapping wing
Full Flow Field Observation and Measurement, Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of China
Author for correspondence (e-mail:
gx_shen55{at}yahoo.com.cn)
Accepted 16 February 2008
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Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Key words: flapping wing, hovering, flow structure, leading-edge vortex (LEV), vortex shedding, electromechanical model, digital stereoscopic particle image velocimetry (DSPIV)
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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; Van Den Berg and
Ellington, 1997; Liu et al.,
1998; Dickinson et al.,
1999; Birch and Dickinson,
2001; Sun and Tang,
2002; Birch et al.,
2004; Wu and Sun,
2004; Bomphrey et al.,
2005). The substructure of this vortical system was reported by
Srygley and Thomas (Srygley and Thomas,
2002), and recently rediscovered
(Lu et al., 2007) and
confirmed (Lu et al., 2006).
Furthermore, the LEV shedding was observed at the outer wing
(Lu et al., 2006), which had
not been reported before. These new discoveries strongly suggested that the
complexity of the LEV system had been underestimated in the earlier
investigations. As a result of the 3-D and unsteady nature of the flapping
wing LEV system, not only the 3-D flow measurement technique but also
systematic study of the different phases within a stroke period are required
to reveal its actual flow structure and evolution. In fact, this knowledge is
of fundamental significance as it serves as a basis for the more intricate
case of dragonfly flight with forewing–hindwing interactions.
As far as we know, there have been only a limited number of studies
addressing the 3-D flow structures and evolution of the flapping wing LEVs.
Liu et al. conducted a computational study of hawkmoth hovering at high
Reynold's number (Rê3000) and presented the 3-D LEV structures
in evolutionary sequence (Liu et al.,
1998), but the vortical structures were represented via
the 3-D streamlines, which could not educe the topological structures of the
vortices and would be problematic when interpreting the unsteady flow field
(Hama, 1962). Recently,
digital stereoscopic PIV (DSPIV), a 2D3C (two-dimensional, three-velocity
components) flow imaging technique involving two cameras positioned at the
stereoscopic configuration (Arroyo and
Greated, 1991), was implemented for measurement of the full flow
field around a flapping model fruitfly wing (Rê100) at various
time steps (Poelma et al.,
2006). Based on the acquired 3-D data, the velocity gradient
tensor (
u) based vortex identification criterion was
applied, which effectively visualized accurately those vortical structures
that could not be obtained from qualitative visualizations (smokes, bubbles or
dyes) and the common quantitative plots (3-D streamlines or iso-vorticity
surfaces). Nevertheless, the focus of that work was the general flow field,
rather than detailed structure of the LEV system. More importantly, the LEV
system at such low Re values would be structurally simpler than that
in the high Re situation (Birch et
al., 2004; Lu et al.,
2006). Since the typical Re of dragonflies is relatively
high, of the order of 1000 (Dudley,
2000), and larger flapping wing vehicles could more likely be
realized, knowledge of the LEV system at high Re is more
desirable.
Here, as an extension of our previous work
(Lu et al., 2006), we reveal
for the first time detailed 3-D structures and evolution of the LEVs on a
flapping wing. Based upon an electromechanical model dragonfly wing flapping
in a water tank (at Re=1624), the DSPIV technique was implemented.
Thanks to the periodic nature of the flows, we were able to take measurements
at different spanwise locations at separated times for a given stroke phase by
employing the phase-lock technique. These 2D3C velocity vectors were then
assembled along the spanwise direction and formed a 3D3C velocity field. The
evolution of the LEV system was examined by applying the above procedure to
three typical stroke phases: (1) at 0.125T (T=stroke
period), when the wing was accelerating, (2) at 0.25T, when the wing
was moving at maximum speed, and (3) at 0.375T, when the wing was
decelerating. Using these 3D3C velocity data, four vortex identification
criteria [|
|-criterion, 
-criterion
(Chong et al., 1990),
Q-criterion (Hunt et al.,
1988) and 
2-criterion
(Jeong and Hussain, 1995)]
were tested to find out the most suitable criterion for reconstructing
accurate vortical structures.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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), which might coexist together with the LEV
sub-structures, rendering it difficult to differentiate between them. Other
geometric factors such as the curvature of the leading edge as well as the
camber might have certain effects on the LEV system, but there are few reports
that systematically address this issue. Hence it would not be sensible to add
such factors, whose effects are not well studied, to the current experimental
system. Mechanical constraints meant that the model wing was mounted on the
tip of the rotational shaft, and the wing base was 46 mm away from the
translational axis, rendering the effective wing length (r)=104 mm
for the model wing (see Fig.
1A, left). The model wing was fabricated from a flat aluminium
sheet (1 mm thick; c-based normalized thickness=3.8%). The wing
surfaces were painted black to inhibit laser reflection in the DSPIV
experiment.
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The flapping kinematics included two degrees of freedom (d.f.): translation
and rotation. As sketched in Fig.
1B, translation is the azimuthal rotation of the wing about the
translational axis oy, and rotation is the supinating/pronating
rotation about the axis oz (located at 1/4 wing chord from the
leading-edge, denoted in Fig.
1A as a thick black line). D and U in
Fig. 1A are two translational
extremes, and they define the horizontal stroke plane and the stroke amplitude
. The instantaneous translational angle 
(t) varied
as the cosine function (Ellington,
1984):
 | (1) |
(t)
[(t)=90–
(t)°, where
(t)=instantaneous angle of attack;
M=the maximal rotational angle, relating with the mid-stroke
angle of attack mid=90–M°] varied
as a simple harmonic function when the wing was undergoing rotation, but
remained constant when the wing was purely translating. The duration of
rotation Tr was fixed at 0.2T, thus the
rotational function in one period was:
 | (2) |
was set to 60°, the same as for the dragonfly
hovering (Norberg, 1975).
Dragonflies use incline stroke plane in hovering flight
(Norberg, 1975), necessitating
an asymmetric kinematic pattern of down- and upstrokes for weight support and
thrust cancellation (Wang,
2000a; Wang,
2004). Thus the angle of attack in downstroke
(mid=60°) is larger than that in upstroke
(mid=30°). However, according to our observations, the
LEVs produced at the same angle of attack but in symmetric and asymmetric
strokes show no detectable difference in structure
(Lu et al., 2006;
Lu et al., 2007). Because the
sub-LEV structures were observed to be more conspicuous at
mid=60° than those at mid=30°
(Lu et al., 2006), we set
mid=60° for the model wing (M=30°).
The stroke frequency n (=1/T) was set to 0.2 Hz so that
Re=1624 [Re=Utipc/
, where
Utip=mean wing-tip speed; =kinematic viscosity
(Ellington, 1984); here, since
Utip=2nR,
Re=2nRc/], within the range of dragonfly hovering
(Dudley, 2000).
The flapping motions were mimicked via a self-designed
electromechanical system introduced previously
(Lu et al., 2006). In this
study, because the down- and upstroke motions were symmetric, only the phases
in the downstroke were measured.
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A 3 mm thick laser-sheet was created by a dual-pulse Nd-Yag laser system
(maximum of 200 mJ pulse–1, LABest, Beijing, China). Two CCD
cameras (1080 pixelsx1920 pixels, Red Lake, San Diego, CA, USA) with the
close-up lenses (Micro Nikkor 105 mm f/2.8, Nikon, Tokyo, Japan) on
Scheimpflug mounts were positioned as an asymmetric angular-displacement
configuration (Coudert et al.,
2000). The left camera (CCD1) was perpendicular to the
laser-sheet, as in the classical 2-D DPIV arrangement, while the right
counterpart (CCD2) viewed obliquely through a water-filled prism with an
effective angle of roughly 40° with respect to the laser-sheet normal (see
Fig. 2A). The water prism was
utilized so that the oblique-viewing camera had a nearly orthogonal
orientation with respect to the liquid–air interface. This method
effectively reduced radial distortions owing to the large off-axis angle
(Prasad and Jensen, 1995).
Coudert et al. (Coudert et al.,
2000) pointed out that the asymmetry of the stereo setup is, in
fact, propitious to the precision of a DSPIV system [measured by the error
ratio: the ratio of the out-of-plane root mean square (r.m.s.) error to the
in-plane r.m.s. error (Lawson and Wu,
1997b)]. The present asymmetric arrangement is expected to achieve
an error ratio <2.5 (Coudert et al.,
2000). Following the proposal of Lawson and Wu
(Lawson and Wu, 1997a), the
aperture of f/16 (f=focal length) was set for both lenses to
have a large depth of focus.
Calibration was carried out based on a 25 mmx40 mm
(horizontalxvertical) rectangle region (the calibration target), which
was marked with a grid of 21x37 filled circles and printed on a plate
mounted on the positioning translation rail. The calibration target was
recorded by both cameras at five 0.5 mm-spaced locations across the
laser-sheet width. Because the recorded images (especially those viewed by the
oblique camera CCD2) were distorted due to the deviation of the lens axis from
the calibration plate normal, they were at first dewarped to have an
orthogonal coordinate (Willert,
1997). Then the calibration coefficients, which would back-project
the left and right pixel-based 2D2C displacements onto the 2D3C physical
displacements, were calculated for the dewarped images using the least-squares
polynomial approach (Soloff et al.,
1997).
After calibration, the water was seeded with hollow glass beads (1–5 µm diameter), and the measurements of the target flow fields performed. Due to the periodic nature of the flows generated in flapping motions, we could measure different spanwise locations at separated times for each stroke phase. This was done based on the phase-lock technique, which relies on two digital delay/pulse generators (DG 535, Stanford Research System Co., Sunnyvale, CA, USA) to ensure synchronization of the laser pair triggering, the image pair recording and the wing motion. A laser-pulse separation of 4 ms was set for the case of 0.125T, and 2 ms for both cases of 0.25T and 0.375T. For each stroke phase, 23 spanwise locations (5 mm-spacings), ranging from the wing-base to the region beyond the wing-tip, were measured. For each spanwise location, 30 periods were sampled, but the first five periods were discarded to avoid the `start-up effect'.
Although the model wing was painted black, the laser reflections were still
strong in the particle images, which would lead to spurious vectors in the
following cross-correlation. Here, they were removed by subtracting an average
background image, which was produced by averaging all frames in the same
sequence so that the particle grayscales were adequately lower than that of
the reflections (Fore et al.,
2005). The reflection-free particle images were then
preconditioned using a simple approach
(Shavit et al., 2006), which
could effectively inhibit the spurious vectors in the cross-correlation. All
preconditioning and post-processing were operated in Matlab. For convenience
of manipulation, we used MatPIV v.1.6.1, an open-source code for Matlab
(Sveen, 2004) to
cross-correlate the image sequences. Multiple-pass with the final
interrogation window of 16 pixelsx16 pixels was used, generating
excellent vector maps. The vector maps of the same sequence were averaged to
generate the mean vector field. Since misalignment between the laser-sheet and
the calibration target was inevitable
(Willert, 1997;
Coudert and Schon, 2001), a
correction based upon the `disparity map'
(Willert, 1997;
Coudert and Schon, 2001) was
performed. Finally, the left and right 2D2C vectors were back-projected to the
2D3C vectors using the calculated calibration coefficients.
For each stroke phase, the 2D3C arrays were assembled along the spanwise
direction (Z) and formed an 111x55x23 matrix. The raw
3D3C vector field is shown in Fig.
2B. It was then interpolated along the Z-direction to
make all dimensions have equal spacing between the grid points. The final size
of the 3-D matrices was 111x55x245. Subsequently, the vorticity
field (), velocity gradient tensor (u) and the
relevant quantities were calculated.
Vortex identification
A proper vortex identification criterion is critical in the current study.
We tested four well known criteria, which are based on the velocity gradient
tensor
u=
ui/xj
(where i and j are the indices; i, j=1, 2, 3), with the acquired 3D3C velocity
field.
|-criterion. || is the norm of
the vorticity vector (=xu). This criterion
identifies a flow region as a vortex when || reaches a
specified threshold. -criterion. is the discriminant of u's
characteristic equation. It defines a region as a vortex if every point in
this region has >0 (Chong et al.,
1990). governs the instantaneous local stream patterns in
a frame that is relatively at rest with respect to the fluid particle. When
>0, u has complex eigenvalues, and the
instantaneous streamlines are locally closed or spiral
(Chong et al., 1990). u. It defines a region as a vortex if every point in this
region has Q>0 (Hunt et al.,
1988). In incompressible flow
(.u=0),
,
where
=0.5(u–uT)
(T denotes the transpose) and
S=0.5(u+uT)are the
asymmetric and symmetric components of u, respectively;
is the Euclid norm of a given tensor A. Q indicates the local
competition between the rotation rate and deformation (or strain) rate, thus
Q>0 means that the local rotational effect dominates
(Hunt et al., 1988). 2-criterion. 2 is the intermediate
eigenvalue of the symmetric tensor 2+S2,
which relates the pressure P with the relation:
2+S2=–1/[(P)]
when discarding the effects of unsteadiness and viscosity
(Jeong and Hussain, 1995).
This criterion defines a region as a vortex if every point in this region has
2<0, since 2<0 implies that the
plane perpendicular to the local vortex axis has the local pressure minimum
(Jeong and Hussain, 1995).
Fig. 3 shows the
visualizations obtained using these four criteria. Data obtained at
0.25T were used for testing because at this instant the vortical
system had the most intricate structure. The values of
||, and Q were normalized by their
maxima (>0), while the value of 2 was normalized by the
minimum (<0). The thresholds of the criteria were selected carefully so
that the isosurfaces described the basic and close topological structures of
the same vortices.
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|-criterion, although
visualizing the general structures, had the disadvantage of also showing the
shear-layers near the wing surface and between the vortices. The
reconstruction of the -criterion showed substantial noise, and the
borders between the target vortices were obscured. Both of the other two
criteria showed the interesting details more clearly, and illustrated nearly
identical structures owing to their mathematical and physical similarities
(Jeong and Hussain, 1995;
Cucitore et al., 1999;
Chakraborty et al., 2005).
Since the Q-value offers more abundant information about the local
flow field, e.g. Q<0 means the local deformation-rate dominates,
we chose it as the chief criterion in this study. Note that the isosurfaces
educed by these criteria are the vortex core structures, but not the vortex
tubes as defined (Saffman,
1992).
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) are provided
in Fig. 4 for comparison. In
Fig. 5, the 3-D vortical
structures are visualized using two levels of Q-isosurfaces. Two
perspectives are provided in evolutionary sequence. The contour slices of
Q and W (the spanwise velocity component) are shown in
Fig. 6 to complement the
detailed interior flow features.
Flow fields in the stroke phase of 0.125T
At this instant, the wing has completed pronation and is accelerating with
a fixed mid-stroke angle of attack (mid=60°).
The DSPIV reconstruction in Fig. 5A shows a general hairpin-like vortical system on the wing, constituted by the LEV, the trailing-edge vortex (TEV) and the wing-tip vortex (WTV). The LEV is, in fact, a collection of four elements: one primary vortex (Lp) and three minor vortices (Lm1, Lm2 and Lm3). The TEV is denoted T1 at this instant because another component of it will be created and shown in the next phase. The WTV, although generally seeming to be a single vortex, has shown a trend to break up into two substructures: W1 and W2. The dye visualization does not show such a hairpin because the dyes were only released at the leading edge.
Both of the DSPIV and dye pictures indicate that there is no LEV structure at the inner part of the wing (Fig. 5A and Fig. 6A). This is mainly due to the low local wing speed, which cannot cause intense flow separation and form a vortex. The DSPIV result shows that at the mid portion of the wing, the primary vortex Lp has been created and is of considerable strength attaching on the leading edge (see Fig. 6A). However, this is not shown in the dye visualization (Fig. 4A), because the region where the Lp stays is clouded by the remaining structures left in the previous stroke stages.
At the outer wing, the minor vortices Lm1 and Lm2 have been shed (see
Fig. 4A,
Fig. 5A,
Fig. 6). Together with the
outboard minor vortex (Lm3) that remains on the leading edge, they constitute
a vortex street. Thus, our previous observation of the outer wing LEV shedding
(Lu et al., 2006) is
confirmed. At this time, Lm1 connects well with Lp and shows no evident
boundary. They were virtually both parts of a single vortex, which links to W1
via Lm1 (Fig. 5A).
Several typical instantaneous 3-D streamlines are plotted and further reveal the flow field features. The white LEV streamlines belong to the LEV and start at roughly the mid wing. They are spiraling out, stop and meet the green WTV streamlines at the wing-tip. The path traveled by the white streamlines includes Lp and Lm1, indicating that there must be substantial spanwise flows within these structures. And this is confirmed in Fig. 6B. Positive W is strong in Lp and Lm1. The maximal value appears at the mid wing and is completely comparable with the mean wing-tip speed.
Flow fields in the stroke phase of 0.25T
At this instant, the translation of the wing reaches maximum speed.
Generally speaking, the hairpin system is further diversified and becomes more
complicated.
The wing acceleration enhances the streamwise (or chordwise) convection, the rate of which is greater at the outer wing due to the spanwise distribution of the wing speed. Consequently, the outer wing portion of the primary vortex Lp is at first peeled from the leading edge, and spreads to its mid and inner wing portions. Meanwhile, also due to the wing acceleration, the flows at the inner wing can be separated intensively, making the formation of the local vortex possible. By contrast, the reverse spanwise flow created by the WTVs pushes the outer wing flows towards the inner wing. The above combined process not only leads to the deviation of Lp from the leading edge, but also the migration of its origin from the mid wing to the wing-base (compare Fig. 4B with Fig. 5B). Despite these spatial changes, Lp is still bound to the wing surface.
From Fig. 5B, we see that Lp
is slightly conical in structure, because at this stage Lp begins to break
down at its tip. Vortex breakdown is a dramatic change at some points of a
vortex, including axial speed drop and vortex core expansion
(Leibovich, 1984). The 3-D
streamlines are plotted to highlight this flow phenomenon: the white
streamlines released near the wing-base are spiraling out, expand and stop
around the breakdown location (at 0.66R)
(Fig. 5B). The dye
visualization does not show violent breakdown of Lp, probably because the dyes
had not reached the breakdown location at this time, or the beginning of the
breakdown did not cause any violent change in the local flow field
(Fig. 4B). Actually, the
current breakdown location is closer to the wing base than those reported in
previous studies (Van Den Berg and
Ellington, 1997; Liu et al.,
1998), which were at 0.75R. In addition, strong spanwise
flow indeed exists in Lp (Fig.
6D), where the peak speed was over twice the mean wing-tip speed,
and twice the previously reported values
(Van Den Berg and Ellington,
1997; Liu et al.,
1998).
At the outer wing, Lm1 and Lm2 are convected downstream with enlarging distance between them (Fig. 5B). The dye visualization clearly demonstrated such a trend (compare Fig. 4A, B), which is also displayed by the high strain-rate regions indicated with negative Q-regions between the vortices in Fig. 6C. By contrast, WTV is completely separated into W1 and W2. Through the above dynamic process, Lm2, W1 and T1 are separated from the initial hairpin, and they form a new sub-hairpin (Fig. 5B).
Lp develops towards the wing base and is static with respect to the wing surface because of its attachment, while Lm1 is convected downstream. These relative movements cause Lm1 to be split from Lp. This phenomenon more clearly in Fig. 5B, right column. Since it is no longer supplemented with vorticity from the wing boundary, Lm1 is diffused dramatically by the viscosity, and becomes structurally unstable (see Fig. 5B, Fig. 6C). By contrast, Lm2 is somewhat strengthened, having been stretched by the reverse spanwise flow (see Fig. 6D) and linked to the breakdown portion of Lp. At this time, the vortex street at the outer wing becomes more conspicuous (see Fig. 5B).
The deviation of Lp leaves certain space at the mid and inner sections of
the leading edge, where the flow keeps on separating and creating vorticity.
Consequently, Lm3 is established at the mid and inner sections of the leading
edge. Eventually, all spanwise segments of Lm3 connect together along the
leading edge (Fig. 5B and
Fig. 6C). As a matter of fact,
the inner wing sections of Lp and Lm3 refer, respectively, to the primary
vortex and minor vortex of the dual-LEV, identified and confirmed in our
previous studies (Lu et al.,
2006; Lu et al.,
2007).
Similarly, Lm3, W2 and T2 form another sub-hairpin behind that constituted by Lm2, W1 and T1 (Fig. 5B). Hence, as far as the whole vortical system is concerned, the shedding at the outer wing represents the shedding of the hairpin vortex.
Another interesting phenomenon is the watershed of spanwise flows
established at the breakdown location of Lp. From
Fig. 6D we see that on its
inner and outer wing sides, the positive and reverse spanwise flows dominate,
respectively. We plot two typical streamlines to highlight the existence of
the reverse spanwise flow. In Fig.
5B, the magenta streamline starts at the wing tip and is spiraling
with a loose structure towards the breakdown location. The yellow streamline
in Lm2 is released at about 0.84R and it also stops around the
breakdown location. In fact the computational study
(Liu et al., 1998) reported
the reverse spanwise flow within the outer wing LEV, which was denoted LEV2 in
that study. However, this structure was created near the end of the downstroke
(Liu et al., 1998), which is
later than the result of our present study.
Flow fields in the stroke phase of 0.375T
At this instant, the wing is decelerating. With the deceleration of the
stroke, the whole vortical system on the wing is considerably dissipated.
At this time, the breakdown of Lp becomes more dramatic, as exhibited by the dye visualization (Fig. 4C) and also supported by the DSPIV result (Fig. 5C). We can see from Fig. 5C (or Fig. 6E) that both the tip of Lp and its streamlines expanded much more dramatically compared to the prior phase of 0.25T. The breakdown region is shed from the wing, making some part of it out of the measurement window. This dramatic change of the tip results in a torch-like structure for Lp (see Fig. 5B). Moreover, the breakdown location moves into 0.57R and further deviates from the leading edge (see Fig. 4C, Fig. 5C and Fig. 6E).
By contrast, there is no concrete structure of Lm1, Lm2 and W2, and W1 is very distant from the wing (see Fig. 5C, right). In addition, Lm3 is considerably weakened and is collapsed into several discrete segments: the outer wing segments are shed; the inner wing segments, although not shed, are detached from the leading edge with a certain distance (Fig. 6F).
Nevertheless, the spanwise flow in Lp does not drop. Instead, its maximum is increased to over three times the mean wing-tip speed. Moreover, the watershed vanishes and the positive spanwise flow again dominates over the wing.
At this time, T2 keeps on growing at the outer wing, while T1 has completely moved out of the measurement window. The orientation of Tr becomes parallel to the wing chord, indicating the downstream movement of T1.
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DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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1000) the LEV system on a flapping wing is, in fact, a
complex collection of four vortical elements: one primary vortex (Lp) and
three minor vortices (Lm1, Lm2 and Lm3), instead of a single conical vortex,
as reported previously (Ellington et al.,
1996; Van Den Berg and
Ellington, 1997; Liu et al.,
1998; Birch et al.,
2004).
We used a model dragonfly wing with high AR and without a curved
leading edge. In our earlier study (Lu et
al., 2006), however, DPIV measurements demonstrated that the
sectional flow structure of the LEV system was not sensitive to AR
and the leading-edge curvature, and the dye visualizations showed similar
evolutionary process of the LEV system. We estimate that in the hovering
condition at high Re, the 3-D flow structure and evolution of the
flapping wing LEV system shown in the present study could, in general, be the
basic pattern. Other geometric factors such as corrugation, camber and twist
were not considered here; these are certainly of interest and should be
studied to see how their effects alter or interact with the basic structures
shown in this study.
Nevertheless, at low Re (100) the LEV system would be
different in structure: the spanwise flow would be weaker
(Birch et al., 2004;
Lu et al., 2006) and exist
behind the LEV region (Poelma et al.,
2006), and the sub LEV structures would not be so dramatic
(Lu et al., 2006). In fast
forward flight, the structure of the LEV system could also be changed: the
flows would become more attached to the wing as the increase of the flight
speed (or advanced ratio) (Sun and Wu,
2003; Wang and Sun,
2005).
Here, it is necessary to clarify two terms related to the flapping wing LEV system.
;
Van Den Berg and Ellington,
1997; Birch et al.,
2004), should refer to Lp. Of the elements of the LEV system, Lp
is strongest and most evident, making it the first element to be discovered
(Ellington et al., 1996). At
low Re, the sub LEV structure was not observed
(Lu et al., 2006), thus the
term `LEV' could refer to the simplex vortical structure attached on the
leading edge (Dickinson et al.,
1999; Birch and Dickinson,
2001; Birch et al.,
2004). )
and confirmed (Lu et al.,
2006) in our previous studies, is constituted by the mid and inner
sections of Lp and Lm3 at mid-stroke. It is virtually a subset of the LEV
system at a certain stroke phase when a wing flaps at certain high
mid and Re.
In our previous work (Lu et al.,
2006), based on the structural similarities, we tried to gain
insight from the realms of delta wings
(Gordnier and Visbal, 2003;
Taylor and Gursul, 2004;
Henning et al., 2005) to
explain the formation of the dual-LEV (Lu
et al., 2006). Admittedly, such an idea has an intrinsic problem.
The delta wings were fixed and the visualizations were conducted when the
flows reached the steady state; the hypotheses of the dual-LEV formation were
made according to the steady-state flow pictures. However, in the case of
flapping wing, the flow field is highly unsteady and has no chance of reaching
a steady state due to the dynamic motions of the wing. The present result
demonstrates that the dual-LEV is a consequence of the dynamic evolution of
the LEV system. In general, its formation is directly related to the movement
of Lp as well as the vortex establishment of Lm3 at the inner wing.
The `stay properties' of the LEV elements on a flapping wing
In our previous study (Lu et al.,
2006), we reported our preliminary results demonstrating that at
different spanwise locations the LEVs have distinct flow behaviors. From the
results of the present study, it appears that these flow behaviors are
specified as the stay properties, which worsen as they approach the wing tip:
(1) at the inner wing, Lp is attached well on the wing; (2) at the mid wing,
Lp breaks down; (3) at the outer wing, Lm1 and Lm2 are shed.
The attachment of the flapping LEV system has been well known for a long
time (Ellington et al., 1996;
Van Den Berg and Ellington,
1997; Liu et al.,
1998; Dickinson et al.,
1999; Birch and Dickinson,
2001; Birch et al.,
2004). The LEV breakdown has also been confirmed
(Liu et al., 1998;
Birch et al., 2004;
Lu et al., 2006;
Lu et al., 2007). The outer
wing LEV shedding has only been reported in a computational study
(Luo and Sun, 2005) and in our
previous paper (Lu et al.,
2006), but is now confirmed by the results of the present study.
The existence of the shedding of the outer wing minor vortices could impact
the spanwise pressure distribution (Luo
and Sun, 2005). Nevertheless, the resultant vortex dynamics should
essentially be unchanged since large part of the force is produced by the
attached primary vortex.
With the current images, we also show that the stay property of the whole
LEV system is worsened as the stroke proceeds. During the wing acceleration,
Lp begins to break down, and Lm1 and Lm2 become more distant from the wing.
During the wing deceleration, the breakdown of Lp is intensified, and Lm1 and
Lm2 are dramatically dissipated. Therefore, the direct acting area of the LEV
system upon the wing is reduced with time during a stroke. Nevertheless, the
vortex dynamics may not necessarily be decreased with the same manner
(Wu and Sun, 2004), since the
primary vortex is the dominating element responsible for the vortex dynamics
on the flapping wing and during the wing acceleration it is being
strengthening.
The spanwise flows
The spanwise flow in the LEV region is a hot topic in the field of flapping
wing as it is considered to be one of the crucial factors responsible for the
stability of the LEV at high Re
(Ellington et al., 1996;
Van Den Berg and Ellington,
1997; Liu et al.,
1998; Birch et al.,
2004). The results of the present study indicate that the spanwise
flows are time-dependent, not only in the magnitude of the velocity but also
in the direction.
At the early stage of the stroke, spanwise flow exists mainly at the mid
and outer sections of the wing (Fig.
6B), in agreement with our earlier dye visualization studies
(Lu et al., 2007) and
supporting the computational results (Liu
et al., 1998). As the wing accelerates, this positive spanwise
flow moves into the inner section of the wing, accompanying the emergence and
growth of the reverse spanwise flow at the outer wing. The competition of
these two opposite spanwise flows eventually causes the breakdown of the
primary vortex and establishes a watershed at the breakdown location (see
Fig. 6D). Liu et al. also
reported a similar phenomenon of diversification of the spanwise flows, but it
appeared at a later phase than our result (Lu et al., 1998). In the real
dragonfly free flight (Thomas et al.,
2004), the non-uniformity of the spanwise flow direction was also
visualized, and was reported to depend on the degree of sideslip. This implies
that the incoming flow introduced in the experiment might play a role in
causing the change of the direction of the spanwise flows. During the wing
deceleration, the positive spanwise flow not only regains control over the
whole wing but is even also strengthened, regardless of the collapse of the
primary vortex (Fig. 6F). This
phenomenon has seldom been reported.
At high Re, a strong positive spanwise flow is always found in the
LEV region (actually in Lp), which is able to `drain' the vorticity and
prevent the overexpansion (instability) of the LEV (Lp)
(Ellington et al., 1996;
Van Den Berg and Ellington,
1997; Birch et al.,
2004). According to the present result, however, the effect of
this vorticity transportation to the stabilities of the LEV elements is
limited. From a certain phase of the stroke, the appearance of the reverse
spanwise flow begins to block further transportation of the Lp vorticity into
the wake. Therefore, the vorticity is accumulated at the tip of Lp, causing
local instability and eventually the breakdown. At the outer wing, though
holding (reverse) spanwise flow, Lm1 and Lm2 are still unable to avoid being
shed. Compared with the hawkmoth studies
(Ellington et al., 1996;
Van Den Berg and Ellington,
1997; Liu et al.,
1998), the current maximal speed of the spanwise flow in the
primary vortex is higher, over twice their reported values. Our preliminary
estimate is that that AR might be a factor. Some studies on free
flight of real insects argued that spanwise flows were not dominant
(Thomas et al., 2004;
Bomphrey et al., 2005). It is
difficult to rule out the effect of incoming flow to the LEV system and the
spanwise flows, however, because it is one major difference between their
studies and the hovering experiments
(Ellington et al., 1996;
Van Den Berg and Ellington,
1997; Liu et al.,
1998; Birch et al.,
2004; Lu et al.,
2006; Lu et al.,
2007).
At low Re values (100), the spanwise flow was detected behind
the LEV region (Poelma et al.,
2006), and the stability mechanism of the LEV structure could be
different (Wang, 2000b;
Birch and Dickinson, 2001;
Poelma et al., 2006).
The LEV systems on flapping wings and sweepback fixed wings
It was found that when Re reaches the order of 1000, the
leading-edge flow structures on the flapping wings are analogous to those on
the fixed delta wings, for example, the conical primary vortex, the intense
spanwise flow, the LEV breakdown and the dual-LEV structure
(Ellington et al., 1996;
Van Den Berg and Ellington,
1997; Liu et al.,
1998; Birch et al.,
2004; Lu et al.,
2006; Lu et al.,
2007).
Undoubtedly, the leading edges are the vorticity-feeding sources for the
LEV systems of both kinds of wings. However, the other flow mechanisms are
totally distinct. The delta wings are static; the sweepback of the wing is the
most significant factor responsible for the flow field behaviors, as it allows
the incoming flow to have a velocity component along the leading edge, which
transports the leading-edge vorticity outward and thus stabilizes the primary
vortex of the LEV system (Wu et al.,
1991). Unlike the delta wing, flapping wings are highly dynamic;
the revolving nature of the flapping motion creates the linear spanwise
distribution of the wing speed, which induces the spanwise pressure gradient,
centrifugal acceleration and the Coriolis acceleration
(Van Den Berg and Ellington,
1997). Either one of these effects or their combination could be
the impetus for the generation of the positive spanwise flow
(Van Den Berg and Ellington,
1997). Also, the dynamic motions of the flapping wings lead to the
time-dependent behaviors of the spanwise flows.
Furthermore, the relation between the strength of the wing-tip effect and
AR is different for flapping wings and fixed wings. According to the
classical fixed wing theory, it is well known that the higher the AR
of a wing, the weaker the wing-tip effect, and the better is the lift
performance. As to a flapping wing, increasing the AR is bound to
enhance the relative wing-tip speed, and this actually reinforces the wing-tip
effect. In this case, the vorticity transportation of Lp would be blocked more
severely. Hence the increase of the AR is virtually detrimental to
the stability of Lp and, further, the aerodynamic performance. Nevertheless,
in the realm of low Re (100), where the spanwise flow is not
striking while attached LEV structure is prominent
(Birch and Dickinson, 2001;
Birch et al., 2004;
Lu et al., 2006), the wing-tip
effect would play different role. Actually, it has been found that the
wing-tip effect could enhance the leading-edge stability by reducing the
effective angle of attack with the induced downwash
(Birch and Dickinson,
2001).
Concluding remarks
In this experimental study, we implemented DSPIV and for the first time
uncovered the detailed 3-D flow structure and evolution of the LEV system on a
flapping wing. It is found that the LEV system is a complex collection of four
vortical elements: one major vortex (Lp) and three minor vortices (Lm1, Lm2
and Lm3). The complexity of the LEV system is also the result of the
diversifications of the spanwise flows and the stay properties of the LEV
elements at different spanwise sections of the wing and at different stages of
the stroke. It is of interest to see how these LEV elements behave with the
forewing–hindwing interactions in the future.
LIST OF ABBREVIATIONS AND SYMBOLS
u
u
u+uT)]
mid
(t)
u's characteristic
equation
Tr
22+S2
M
(t)
(t)
xu)
u–uT)]
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Acknowledgments |
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Footnotes |
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References |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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