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51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
07 - 10 January 2013, Grapevine (Dallas/Ft. Worth Region), Texas
AIAA 2013-0952
Effect of the pre-existing camber on fluid–structure interaction
of cicada wings
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Hu Dai, ∗ Haoxiang Luo, † Jialei Song, ‡ James F. Doyle §
Insect wings consist of a membranous structure with vein reinforcement, which gives the wings
necessary stiffness and meanwhile proper flexibility to enhance their aerodynamic performance.
Many insect wings also have a pre-existing camber that not only improves the spanwise stiffness
of the wings but also leads to asymmetry in the wing flexibility. To study the effect of such camber on
the dynamic deformation of the flexible wings and investigate the aerodynamic consequence, we combine direction numerical simulation of the unsteady flow and a finite-element model of cicada wing
whose geometric and structural properties are based on the experimental measurement in our lab.
The result shows that the camber increases the rigidity of the wing for the load from the ventral side
(i.e., the downstroke) but maintain the wing flexibility for the load from the dorsal side (i.e., the upstroke). Furthermore, the asymmetry introduced by the pre-existing camber significantly enhances
lift production.
I. Introduction
The membranous wings of insects typically experience significant deformation during flight 1,2 . Such
deformation is caused by a combination of the wing inertia and the pressure exerted on the wing surface by
the surrounding air. Several studies in recent years using simplified flapping wing models have suggested
that the wing flexibility may significantly improve the aerodynamic performance of the wing. For example,
Heathcote & Gursul 3 experimentally investigated the effect of chordwise deformation on thrust generation
and power consumption of an elastic foil heaving periodically in water flow. In their study, the chordwise
flexibility was found to increase the thrust efficiency compared to the rigid foil, and in the presence of the free
stream, the thrust and power coefficients strongly depend on the Strouhal number defined with the heaving
amplitude of the leading edge. Zhu 4 used a boundary-element method to solve the potential flow assumed
for a flapping foil while incorporating vortex shedding in the wake. His result on the chordwise flexibility
driven by the fluid force is in general consistent with the conclusion by Heathcote & Gursul 3 . In another
work, Dai et al. 5 used a rectangular sheet, reinforced at the leading edge, as a model to investigate the role
of the wing inertia relative to the aerodynamic pressure in hovering flight. In their work, the ratio between
the wing inertia and the dynamic pressure is represented by a mass ratio, and the flexibility of the sheet is
represented by the ratio between the flapping frequency and the natural frequency of the sheet. Their study
shows that within a range of wing flexibility, the dynamic deformation of the wing significantly increase the
lift production of the wing and meanwhile enhances the lift efficiency. Furthermore, at a lower mass ratio,
the aerodynamic force is able to sustain the wing deformation caused by the inertial force during the wing
∗ Graduate
student, Department of Mechanical Engineering, Vanderbilt University, 2301 Vanderbilt Pl., Nashville, TN 37235
Professor, Associate member of AIAA, Department of Mechanical Engineering, Vanderbilt University, 2301 Vanderbilt
Pl., Nashville, TN 37235
‡ Graduate student, Department of Mechanical Engineering, Vanderbilt University, 2301 Vanderbilt Pl., Nashville, TN 37235
§ Professor, Department of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-2045
† Assistant
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Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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reversal, and the aerodynamic efficiency is further increased as compared to the wing at a higher mass ratio.
Despite the extremely simplified models for representations of insect wings, there has been little more
direct evidence that shows the benefit of the wing flexibility on the aerodynamic performances of the wings.
Note that insect wings have more complicated kinematics than simple translation and pitching motions, and
the wing structures are anisotropic and inhomogeneous. The deformation patterns are in general three dimensional. A direct simulation of the insects’ flexible wings would provide more definitive answers to the
questions regarding the role of wing deformation. Recently, Young et al. 6 performed a full-body numerical simulation of the forward flight of locusts by incorporating realistic wing kinematics reconstructed from
high-speed imaging. For comparison, they also performed flight simulations based the modified wing kinematics by removing the camber and spanwise twist from the full-fidelity wing motion. Their result shows that
the wing deformation considerably reduces the power required to generate lift. However, one limitation in
their study is that the wing deformation is specified in the computational fluid dynamics. The study therefore
cannot tell us the cause of the wing deformation. Neither does it allow us to investigate the effect of the structural/aerodynamic parameters that determine the dynamic behavior of the wing. For these reasons, a fully
coupled fluid-structure-interaction (FSI) simulation would be more preferable. To our knowledge, previously
there has been little study in this regard.
Recently we have developed a high-fidelity model to simulate the fluid-structure interaction of the cicada
forewing 7. In this model, a finite-element representation of the wing is coupled with an immersed-boundary
method for the flow. The elastic properties of the wing are based on experimental measurement, and the actuation kinematics is reconstructed from high-speed imaging of the real insect. The model has been validated
by comparing the simulated wing deformation with that observed from the imaging experiment.
A particular question we would like to address in this work is the effect of the pre-existing camber on
the wing deformation and on the aerodynamic performance of the flexible wing. As seen in Fig. 1, many
insect wings have a pre-existing camber that is convex on the dorsal side but concave on the ventral side. It
is expected such a curvature would enhance the spanwise stiffness of the wing. Such spanwise stiffness is
much needed as the wings bear most of their load in the form of the moment about the longitudinal axis of
the body. However, the camber also introduce dorsal-ventral asymmetry to the wing structure, and its effect
on the aerodynamics and on the wing deformation is not yet clear.
In the following sections, we will describe the modeling approach for the elastic wing and briefly sum-
Figure 1. A cicada whose forewing displays a pre-existing camber.
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marize the numerical method. The result and discussions will be given in the end.
II. Modeling approach
A.
Wing structure and material properties
Body weight
0.185 g
Body length
24 mm
Forewing length
3 cm
Forewing chord length
1 cm
Flapping frequency
25 Hz
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Table 1. Morphological data of the cicada used in this study
The 13-year species of magicicada emerged in large amount in middle Tennessee around May, 2011,
which provided ample samples for our study. The morphological data of the insect are measured and listed
in Table 1. High quality images of the forewing were taken, and the vein network of the wing was traced out
in Matlab as shown in Fig. 2. The veins are divided into several groups based on their hierarchical levels and
the visual inspection of their diameters.
The wing geometry was then reconstructed in a mesh generation tool and discretized into linear segments
(for veins) and triangular elements (for wing surface).
We measured the mass distribution of the wing by segmenting the forewing of the cicada into four portions: A – tip half of the wing surface, B – base half of the wing surface, C – base portion of the leading edge
spar, and D – tip portion of the leading edge spar. The areas of parts A and B are calculated in Matlab, and
their masses are measured using a microbalance with accuracy of 0.1 mg. Then the area density of each part,
ρs h, is calculated. For parts C and D, 10 samples of 10 mm in length are weighed together and the average
mass is obtained. Then the linear density of the spar is calculated. The data are listed in Table 2.
For the wing stiffness, each group of the veins as shown in Fig. 2 are measured, and average data were
taken and are listed in Table 3.
High quality images of the cicada wing were taken, and the vein network of the wing was traced out
Figure 2. Wing structure reconstructed from high-quality images where different color represents the manually specified material group of the vein.
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Wing portion
Mass (mg)
Area or linear density
A
1.09
0.9
B
2.50
2.6
C (1 cm base portion)
1.7
1.7
D (1 cm tip portion)
3.5
0.35
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Table 2. Mass and density of different portions of the forewing. A is the near-tip half portion, B is the near-root half portion and
C is the leading edge. The unit of the area density is mg/cm2 , and the unit for the linear density is mg/cm.
Vein level
1
2
3
4
5
6
7
Membrane
Total mass
EI (104 mg·cm3 /s2 )
275.042
66.000
10.500
10.500
4.300
2.500
0.370
4.9
GJ(104 mg·cm3 /s2 )
180.760
40.000
8.400
8.400
3.440
2.000
0.296
–
Linear density (mg/cm)
1.33
0.36
0.29
0.29
0.17
0.067
0.023
2
0.40 mg/cm (surface density)
6.4 mg
Table 3. Stiffness and mass of veins and membrane in the wing structure.
in Matlab. The wing geometry was then reconstructed in a mesh generation tool and discretized into linear
segments (for veins) and triangular elements (for wing surface). We also measured the mass distribution
(Table 2) and stiffness of the major structural elements including the membranous surface and several hierarchical levels of veins. These data are fed into the finite-element model of the wing, which is shown in
Fig. 2.
B.
Specifying the pre-existing camber
The unstructured mesh of the cicada forewing is shown in Fig. 3(a) The curved surface of the wing is measured by putting the wing on a flat surface and then probing the surface. The contour is shown in Fig. 3(b)
To create a similar contour in the wing model, the surface height was specified manually after the mesh was
created. A simplified prescription of elliptical contours is used to replace the measured contour to produce
a smoother distribution, as shown in figure 3(c). The maximum surface height is around 10% of the chord
length.
C.
Reconstructing the wing kinematics
In a filming process a cicada is tethered and stimulated to flap wings. A high-speed camera is used to
recording the wing motion at 1000 frames per second. The time series of images are then used to reconstruct
the wing actuation and track the wing deformation. Markers are labeled on the wing surface to enable tracing
later. Three markers near the root are used for reconstruction of the kinematics at the root, one marker on the
root, one on the leading edge, and one on the trailing edge. The latter two points on the leading and trailing
edge are assumed to have rigid connection with the root point. Since only one camera is used, an algorithm
is needed to reconstruct the 3D coordinates of the points tracked from the 2D coordinates. This is done by
measuring the actual distance between the three markers. As illustrated in Fig. 4(a), given two coordinates x
and z extracted from a 2D image, the third coordinate y of any point A in space can be found out according
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to
y=
p
l2 − x2 − z 2
using the actual distance between A and the root, AO = l.
As long as the coordinates of the three points are known (one of them is the root and is assumed to be the
origin and the pivot point), the three angles, the stroke angle, the deviation angle, and the pitch angle of the
wing can be obtained. As shown in Fig. 4(b), the stroke plane is defined by the plane spanned by two extreme
positions of the leading edge, OH and OL. The position of leading edge, OA, is first projected to OB in the
stroke plane, where B is calculated by
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−−→ −→
−→
OB = OA − (OA · n)n
. Then the angle α between OB and OH, defined as the stroke angle, is calculated as
−−→ −−→
cos α = OB · OH/(|OB||OH|)
, and the angle θ between OB and OA, defined as deviation angle, is calculated as
−→ −−→
cos θ = OA · OB/(|OA||OB|)
. In addition, the pitch angle is introduced to define self-rotation of the wing around the leading edge. The
reconstructed angle are shown in Fig. 4(c).
(a)
3
2.5
2
0.05
0
1.5
0.2
0
1
−0.2
−0.4
0.5
−0.6
0
(c)
(d)
Figure 3. (a) Finite-element mesh of the cicada forewing. (b) Surface contour measured from the real wing. Simple elliptical
contours are used to specify the camber distribution in current wing model.
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(a)
(b)
Z
A’ x
y
A
l
O
X
Y
(c)
Angles (deg.)
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z
Stroke
150
Deviation
100
Pitch
50
0
−50
0
1
2
3
4
5
t/T
Figure 4. (a) A sketch showing reconstruction of 3D coordinates from 2D data from images used for reconstruction of the wingroot actuation. (b) Definition of the angles used to describe the wing actuation. (c) Reconstructed wing-root kinematics as defined
by the stroke, deviation, and pitch angles.
III. Numerical method for the flow–structure interaction
The numerical method of computing the three-dimensional fluid–structure interaction (FSI) is described
in our previous publications and recent work 8,9 . Only a brief summary is given here.
The Newtonian fluid is governed by the viscous incompressible Navier–Stokes equation and the continuity
equation
∂vi ∂vj vi
1 ∂p
∂ 2 vi
=−
+ν 2,
+
∂t
∂xj
ρ ∂xi
∂xj
∂vi
= 0,
∂xi
(1)
where vi or v is the velocity, ρ and ν are the fluid density and viscosity, respectively, and p is the pressure.
No-slip and no-penetration conditions for the fluid are imposed on the plate surface, and the plate is subject
to the hydrodynamic traction differential f = σ (+) − σ (−) · n where σ is the hydrodynamic stress tensor,
n is the surface normal, and plus/minus denote the two sides of the plate.
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The computational fluid–structure interaction is achieved by coupling a Cartesian-grid based immersedboundary flow solver and a nonlinear finite-element structural solver. Both programs were originally standalone in-house codes and have been validated extensively.
The structural solver is the finite-element package for solid mechanics, NonStaD (standing for nonlinear
analysis of statics and dynamics), developed by Professor Doyle at Purdue University. NonStaD is designed
specially for thin-walled structures consisting of frames, membranes, plates, and shells. The software has the
capability of handling large displacements and large rotations, and the constitutive laws include both elasticity
and plasticity. The large-displacement and small-strain deformation in the structural solver is handled using
the corotational scheme. That is, a local coordinate system is envisioned as moving with each discrete element, and, relative to this coordinate system, the element behaves linearly. Consequently, the nonlinearity of
the problem is the result of the coordinate transformation. The dynamical system representing the structural
vibration is obtained by assembling the equations for all the elements,
[M ]{ü} + [C]{u̇} = {P } − {F },
(2)
where [M ] and [C] are the structural mass and damping matrices, {P } is the external force vector incorporating the forces from the fluid in contact with the structure, and {F } is the deformation-dependent vector of
element nodal forces. The time stepping is achieved using an incremental/iterative strategy.
The coupling between the fluid and solid dynamics are achieved by exchanging the boundary displacement, velocity, and traction between the two solvers and performing iteration until convergence is reached.
IV. Results and discussion
A.
Static tests
We first perform a static-load test to examine the effect of camber on the stiffness symmetry of the wing.
Using the finite-element model, we apply a constant force on a chosen point on the wing surface from either
ventral or dorsal side. The force is 60 mg, about one-third of the total mass of the insect. Two positions are
chosen as the loading point, as indicated in Fig. 5 by F1 and F2. The displacement of point C, a point on
the trailing edge obtained from the simulation is used for measurement. Both the cambered wing and the
flat (no-camber) wing are tested. Results are given in Table 4. It’s found that for the cambered wing, the
deformation caused by a ventral force is significantly lower than that by a dorsal force. For the no-camber
wing, symmetric deformation is seen from the table, as expected. Furthermore, the no-camber wing has
greater deformation than the cambered wing when the load is on the ventral side. Therefore, the pre-existing
camber strengthens the flexural stiffness of the wing mainly for ventral loads. The camber thus introduces an
asymmetric stiffness to the wing, which is important for the dynamic deformation of the wing.
Load applied
F1 = 60 mg
F2 = 60 mg
Cambered wing
ventral load dorsal load
0.28
-0.35
0.29
-0.47
No-camber wing
ventral load dorsal load
0.36
-0.36
0.51
-0.51
Table 4. Displacement measured at point C in the static-load tests. The unit of displacement is cm.
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A
O
A
F1
F2
Dorsal O
D
Ventral
C
B
B
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C
Figure 5. The static load tests of the wing where force is applied normal to the wing surface.
B.
Dynamic deformation in vacuum
We then compare the dynamics of the wing with and without camber when the fluid forces are absent.
Fig. 6(a) shows a sequence of the wing shape during different phases during downstroke and upstroke. It
can be seen that without camber, the wing deformation is greatly increased during downstroke. To better see
the difference, we shown in Fig. 7(a) the chord series at 3/4 wing span position, where substantial chordwise deformation can be observed. These figures show that during the downstroke, the no-camber wing has
greater spanwise deformation, wing twist, as well as chordwise deformation. However, during upstroke,
similar deformation is observed between the cambered and no-camber wing.
To further test the effect of the camber, we manually increase the bending stiffness of the leading edge
spar of the no-camber wing by 10 times so that the no-camber wing would have the similar spanwise bending
stiffness during downstroke. Fig. 6(b) shows the comparison between the reinforced no-camber wing and
the cambered wing. Fig. 7(b) shows the wing chord at the 3/4 wing span position. These figures show that
during the downstroke, the reinforced wing has similar spanwise deformation as the cambered wing, and the
reinforced wing has more chordwise deformation and wing twist. During upstroke, the cambered wing now
more spanwise deformation and wing twist.
From these dynamic tests, we see that the camber mainly increases the ventral stiffness of the wing during
downstroke but maintain the spanwise and chordwise flexibility during the upstroke.
C.
Interaction with the flow
We further performed the fluid–structure interaction of the wing with the flow. The fluid is otherwise quiescent (zero free stream velocity). Fig. 8 shows the deformation pattern of the wing as well as the chord
pattern. In this test, the reinforced no-camber wing is chosen to compare with the cambered wing. This is
because the wing with neither camber or the reinforcement produces significantly less lift than the cambered
wing, and using the reinforced wing would allow us the examine the effect of chordwise deformation and the
deformation during upstroke.
Consistent with the dynamic test in vacuum, the FSI simulation shows that during downstroke, the cambered wing is able to maintain the strong stiffness of the wing for both spanwise and chordwise directions.
During upstroke, the camber allows the flexibility of the wing for both spanwise and chordwise deformations.
A wing without camber has different aerodynamic performance by varying the aerodynamic forces and
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(b)
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(a)
Figure 6. Instantaneous deformation pattern of the wing in the dynamic test in vacuum. (a) The cambered wing (orange) and
the no-camber wing (green); (b) the cambered wing (orange) and the no-camber wing with leading edge reinforcement (green).
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(a)
(b)
ok e
ok e
pla
pla
ne
ne
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S tr
S tr
Figure 7. Instantaneous wing chord pattern of the wing at 3/4 span in the dynamic test in vacuum. (a) The cambered wing (red)
and the no-camber wing (black); (b) the cambered wing (red) and the no-camber wing with leading edge reinforcement (black).
changing the power consumption. As shown in Fig. 9(a) the lift force is significantly reduced due to the
removal of camber which is beneficial to capturing the fluid thus producing higher lift during downstroke.
As shown in (b), thrust is also reduced during downstroke. However, the camber somewhat increases the
thrust during upstroke. Table 5 shows the averaged lift, thrust and power data. Overall, by introducing the
camber the average lift is increased by 20% and thrust is reduced by 10%. The power is slightly increased by
the camber, from 23.4 to 25.8×10−4W. Note that the no-camber wing has been reinforced in this test. This
comparison suggests the pre-existing camber is more likely to serve the lift enhancement as compared to a
wing with similar stiffness due to the lead edge reinforcement.
Cambered wing
Reinforced no-camber wing
Thrust(mg)
67
74
Lift(mg)
54
44
Power(10−4W)
25.8
23.4
Table 5. Mean thrust and lift forces and power for the two wings.
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(b)
e
lan
ep
ok
Str
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(a)
Figure 8. (a) Deformation pattern of the wing with camber (orange) and the reinforced no-camber wing (green) in the FSI
simulation. (b) Chord pattern at 3/4 wing span of the wing with camber (red) and the reinforced no-camber wing (black) in the
FSI simulation.
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Lift(10mg)
30
Camber wing
20
No−camber wing
10
0
−10
−20
2
2.5
3
t/T
3.5
4
2.5
3
t/T
3.5
4
2.5
3
t/T
3.5
4
(b)
Thrust(10mg)
30
20
10
0
−10
2
(c)
Power(10−4W)
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(a)
100
50
0
2
Figure 9. History of lift (a), thrust (b) and power (c) shown for two flapping cycles for the cambered wing and the reinforced
no-camber wing.
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To conclude, our computational model of the cicada forewing shows that the pre-existing camber in the
wing surface mainly increases the wing stiffness during downstroke for lift production. Meanwhile, the camber maintains the wing flexibility during upstroke so that significant spanwise and chordwise deformations
are observed.
Acknowledgment This research was supported by the NSF (No. CBET-0954381). The computing resources
were provided by the NSF XSEDE and the Vanderbilt ACCRE.
References
Downloaded by UNIVERSITY OF ADELAIDE on October 29, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2013-952
1 Wootton,
R., “Support and deformability in insect wings,” Journal of Zoology, Vol. 193, No. 4, 1981, pp. 447–468.
S. and Daniel, T., “Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending,” J. Exp. Biol.,
Vol. 206, No. 17, 2003, pp. 2989–2997.
3 Heathcote, S. and Gursul, I., “Flexible flapping airfoil propulsion at low Reynolds numbers,” AIAA J., Vol. 45, No. 5, 2007,
pp. 1066–1079.
4 Zhu, Q., “Numerical simulation of a flapping foil with chordwise or spanwise flexibility,” AIAA J., Vol. 45, No. 10, 2007, pp. 2448–
2457.
5 Dai, H., Luo, H., and Doyle, J., “Dynamic pitching of an elastic rectangular wing in hovering motion,” Journal of Fluid Mechanics,
Vol. 693, 2012, pp. 473–499.
6 Young, J., Walker, S., Bomphrey, R., Taylor, G., and Thomas, A., “Details of Insect Wing Design and Deformation Enhance
Aerodynamic Function and Flight Efficiency,” Science, Vol. 325, No. 5947, 2009, pp. 1549.
7 Luo, H., Dai., H., Das, S. M. A., Song, J., and Doyle, J. F., “Toward high-fidelity modeling of the fluid–structure interaction for
insect wings,” AIAA Paper 2012-1212, 2012.
8 Luo, H., Yin, B., Dai., H., and Doyle, J. F., “A 3D computational study of the flow–structure interaction in flapping flight,” AIAA
Paper 2010-556, 2010, 2010-556.
9 Luo, H., Dai, H., Ferreira de Sousa, P., and Yin, B., “On the numerical oscillation of the direct-forcing immersed-boundary method
for moving boundaries,” Comput. Fluids, Vol. 56, 2012, pp. 61–76.
2 Combes,
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