Introduction
With the requirement of military mission and civil application, due to the ability of
executing the mission in the so called D3 (dull, dirty and dangerous) environments, micro
air vehicle (MAV) has attracted more and more attention. Usually, MAV needs to have
hovering capability and high maneuverability, which is impossible by employing the
conventional fixed wing. So, flapping wing and revolving wing are two popular ways
which is being considered for building MAVs.
Insects have been survived in the earth for about 350 million years, they are the most
agile and maneuverable creatures for their size. Unlike birds, the wing has no muscles
attached to bones and is controlled by one joint at the root, which in one hand makes the
wing very light, in the other hand makes it easier to reconstruct. Therefore, insect flight
renders a feasible way for scientists and engineer to mimic. In order to build an efficiency
machine, people have been trying to discover the underlying mechanism of insect flight
since . Weis-Fogh discovered the clap-and-fling mechanism, which causes large
circulation and generates considerably large lift on the wing. Ellington pointed out that
unsteady effects are pronounced during hovering flight and provided morphological and
kinematics data for a variety of insects. Later, Ellington revealed the importance of
leading edge vortex in lift generation. Dickinson and coworkers employed experimental
setup to identify three distinct mechanisms during insect flight: delayed stall, rotational
circulation and wake capture. Mittal used a CFD model to study the wing-wing
interaction in four-winged insects. Sun calculated aerodynamic derivatives and analyzed
the dynamic flight stability and controllability of different insect models. Based on the
experimental kinematical analysis of a hovering hawkmoth, Manduca sexta, Liu and his
coworker constructed the wing kinematics in terms of time varying positional angle,
rotational angle and the angle of attack to develop models for CFD analysis. Then they
used the in-house computational fluid dynamics solver for multi-body object to simulate
the hawkmoth in hover. They provided the flow structure as well as force and power
generation. However, there is no 3D experimental unsteady force data for validation.
Also, the model wing they employed is rigid, which is not the case for hawkmoth and the
wing flexibility may play an important role. A number of researchers have pointed out
that the flexible wings may gain some benefits in terms of the generation of aerodynamic
mechanisms of lift/thrust production against rigid wings. Vanella investigated the
influence of flexibility on the aerodynamic performance of a hovering wing using twodimensional, two-link model and reached the calculation that flexibility can enhance the
aerodynamic performance. Miller and Peskin used immersed boundary method (IBM) to
explore the importance of wing flexibility during 2D clap-and-fling. Due to the fact that
insects are small, fast moving and flap their wings at high frequencies, it is difficult to
obtain the detailed 3D wing kinematics with a high level of detail, for example, wing
flexibility. In this work, we measured 3-D kinematics for the flexible wings and employ it
as input into our CFD solver.
3D flexible wing kinematics
As mentioned above, we managed to obtain the 3D kinematics of the flexible wings of a
hawkmoth, Manduca Sexta, in hover. The model is based on the experiments of Hedrick
and Daniel (2006), who have obtained high-resolution videos of this insect in three views.
Figure 1 shows the setup which provided quantitative, high-speed videogrammerty of a
hawkmoth hovering in front of an artificial flower. Fig. 2 shows the sample frames from
high speed video recordings in lateral view. Here is how we reconstructed the flexible
wings and body in mesh, which is suitable for input into our CFD solver. First, Gridgen is
used to reconstruct the hawkmoth wing in mesh with zero thickness. Considering that the
movement of fore-and-hind wing is nearly analogous, the fore wing and hind wing are
modeled as one wing instead. Next, the 3D modeling, animation, visual effects, and
rendering software Autodesk® Maya® is used to create a “animation” of the moth in
flight where the wing kinematics are matched very closely to 16 different instances in the
flapping cycle as observed in the high-speed videos in three views. Fig. 3 shows the
match between images and our model at one instantaneous time. It should be noted that
the wing deformation can not be neglected and we attempt to recreate the deformation
observed in the wings. This animation is then interpolated in time with a cubic spline to
produce a high frame-rate animation suitable for input into the CFD solver. The
reconstruction of body is relatively straightforward. The geometry of the moth body is
constructed from a high-resolution (0.005” precision) NextEngine™ laser scanner, as is
shown in Fig. ?
Fig. 1. High-speed insect flight
videogrammetry setup. (needs to be
changed)
Fig. 2. Sample frames from high speed
video recordings in lateral view including
(a) early downstroke, (b) end of
downstroke, (c) mid-upstroke and (d) end
of upstroke.
Fig. 3. Comparison of our model with three videos at one instantaneous time.
Computational Setup
Fig. 4 shows the constructed realistic wing-body model, which is immersed in the 3D
non-uniform cartesian grid. The wings are modeled as deforming membranes while the
body is treated as rigid body. The domain size normalized by mean chord length c is 25 X
20 X 25. The coordinate directions is also shown in Fig. 4 as follows: X direction is in the
horizontal, +X is toward the moth’s back; Z is also in the horizontal, +Z is toward the
moth’s left; Y is in the vertical, +Y points upward. As is shown in Fig. 4, the grid is
designed to provide high resolution in the region around body (region I in Fig. 4(b)) and
the wake region (region II in Fig. 4(b)) in order to capture the detailed vortex structure. It
should be noted that during the flapping flight, the moth wings generate a pair of vortex
rings below the wings. In this case, we keep the resolution in Y direction relatively high
comparing with X and Z direction. Fig. 5 shows the boundary conditions: on all the
boundaries besides bottom, we apply a far-field boundary condition which amounts to
specifying the streamwise velocity component to zero and setting the normal gradients of
the other velocity components to zero. During the flapping flight, the moth wings
generate a downwash flow, which will be shown later. So, on the bottom boundary, a
convective boundary condition which allows the vortex structures to exit the boundary
without any spurious reflections is applied (Dong et al. 2006).
Fig. 4. The constructed realistic wing-body model immersed in the 3D non-uniform
cartesian grid.
Fig. 5. Boundary condition in our
simulation.
Reynolds number and grid independence study
Triantafyllou et al and Dong et al have pointed out that the Reynolds number has a
relatively weak effect on the hydrodynamics of the flapping foils at low Re number. In
this paper, we first examine the Reynolds number effects on the hovering moth
simulation. In order to assess the effect of Re number, two simulations are conducted:
one is Re number 1000 with a grid size of 256X256X192 and the other one is Re number
318 with a grid size of 128X128X128. Both cases provide good resolution of the flow
structure. It should be noted that for Re number 1000 case, in order to save the
computational time, only one wing is used. The reason is based on the fact that body
produces significantly less force than the wings and that the existence of body doesn’t
affect the force produced by the wings. Fig. 6 shows the instantaneous vertical and
horizontal forces produced by the wings and body. The average forces are shown in table
1. Both the figures and table clearly show that the forces produced by the body are
significantly smaller than that produced by the wings. Also, the body has no effects on
the force produced by the wings, which can be shown in Fig. 7. Fig. 7 shows the lift and
drag (thrust) comparison between the two simulations of different Reynolds number. The
difference is very small for both lift and drag (thrust) force. Table 2 shows the averaged
vertical and horizontal forces during one flapping cycle for the two cases. The average
lift is sufficient to support the body weight, and the average drag (thrust) is almost zero
(0.3% of the average lift), which is reasonable for hovering flight. Fig. 8 shows the vortex
core generated at the middle of downstroke for the two cases. We can notice that even the
vortex structure is almost the same for different Reynolds number cases. Considering the
weak effect of Reynolds number and the current computation ability, there is no need to
run a simulation with Re number to be the same as the actual hovering moth. Also, a
study has been carried out to assess the effect of grid resolution on the flow structure and
aerodynamic forces. The grid refinement study is carried out by decreasing the grid
number in the refined region around the wings and body by 20% in all three directions
simultaneously while keeping all the other parameters unchanged. Table 2 shows the
comparison of average lift and drag (thrust) during three cycles between the three
different cases. The differences between the three cases are small, which clearly promises
the fidelity and accuracy of the current simulations.
Fig. 6. Instantaneous vertical and horizontal forces from the wings and body.
Body
Left wing
Right wing
Lift
8.59E-05
7.79E-03
7.81E-03
Drag
2.37E-05
-3.68E-04
-1.23E-04
Table 1. Average forces during one flapping cycle produced by body and wings.
Re=1000 (fine mesh)
Average lift (mN)
Average drag(mN)
Re=1000 (coarse mesh)
Re=318
1.57E-02
1.54E-02
1.55E-02
-4.68E-04
-5.71E-04
-5.20E-04
Table 2. Comparison of average lift and drag (thrust) forces during one flapping cycle.
Fig. 7. The lift and drag (thrust) comparison between numerical simulations.
Fig. 8. The vortex core generated at the
middle of downstroke for different
Reynolds number cases. (Delete the
vortex from the near wing)
In the following paper, if not specified, all the results are from the simulation with the Re
number equal to 1000, which is about one-fifth of that mentioned above for the actual
moth. And for Reynolds number 1000 case, the grid size is 256X256X192 which is
around 12.6 million grid points. The simulations were carried out on Kraken with 256
AMD 2.6 GHz Istanbul-6 processors. It takes around 322560 CPU hours to get a
simulation of three flapping cycles when the flow has reached a stationary state.
Flow structure
Fig. 9. Vortex contour on the plane at 75% of the wing length at seven different times.
(a). Wing trajectory during one flapping cycle. (b). Lift curve during one flapping cycle.
The vortex plot contour on the plane which is at 75% of wing length at seven typical
moments over a flapping cycle is illustrated in Fig. 9. Fig. 10 shows the corresponding
absolute iso-vorticity surfaces around the wing of the hovering moth. During the first half
of downstroke (from 1 to 2), a strong leading edge vortex which covers a large part the
wing surface is generated. Together with LEV, a tip vortex (TE) and a trailing edge
vortex (TEV) are observed to wrap around the wing.
(1)
(2)
Fig. 10. Absolute iso-vorticity surfaces
around the wing of the hovering moth
Fig. 11. Average force contour during one flapping cycle. Left: vertical force; Right:
horizontal force.
Fig. 12. Average force contour during downstroke. Left: vertical force; Right: horizontal
force.
Fig. 13. Average force contour during upstroke. Left: vertical force; Right: horizontal
force.
Fig. 14. Surface streamline for shear force. Left: average shear force during downstorke.
Right: average shear force during upstroke.
Top: shear force contour at the middle of downstroke. Middle: shear force perpendicular
to the leading edge on the wing plane. Below: shear force parallel to the leading edge.
Evaluation of aerodynamic forces, torques
Liu and coworkers also conducted a CFD study for the unsteady flows about a
realistic body-wing model and the force-generation in the flapping flight of hawkmoth
hovering. Their model employ rigid wings and the kinematics are based on Ellington’s
data. Fig. shows that, in both cases, the average lift is comparable with the body weight.
However, there is a big difference between flexible wing and Liu’s simulation.
Fig. 15 shows the instantaneous vertical and horizontal force comparison between
numerical and experimental results. It should be noted that the computational forces are
obtained by averaging the force in three flapping cycles. Due to the lack of detail of the
flow produced by the hawkmoth in hover, the force generated by the insect can otherwise
be determined from its motion. Knowing the mass properties of wing, head, thorax and
abdomen, it is possible to estimate the center-of-mass (CoM) of the moth and track this
CoM using direct linear transformation (DLT). By differentiating the displacement twice
in time, one can determine the acceleration of the CoM and the lift force generated by the
insect can be extracted from this. Our model is based on cyclical repetition of the same
flapping stroke, however, the videos we use in the current research contains the
information for many cycles. The error bar shows the cycle to cycle variation during
different flapping cycles. The plot confirms that the computational vertical force agrees
extremely well with the experimental result. For the horizontal forces, although the
magnitude doesn’t match as well as vertical force, the trend also matches each other very
well and the numerical result falls in the error bar during most of the flapping cycle. It
can be noticed that the peak value of vertical force during downstroke is significantly
higher than that during upstroke due to the strong leading edge vortex. And there are two
peaks of lift during upstroke. The wings produce drag force during downstroke and thrust
force during upstroke. The drag and thrust cancel each other during one flapping cycle
for the hawkmoth in hover. It can be noted that the magnitude of the maximum vertical,
horizontal and sideslip force is the same, which may provide the insect with high
maneuverability.
Fig. 15. The instantaneous vertical and horizontal force comparison between numerical
and experimental results. Left: vertical force; Right: horizontal force.
Fig. 16 shows the vertical (lift) force, horizontal (drag or thrust) force and sideslip force
generate by the two wings. Note that the Reynolds number is set to be 318. The sideslip
forces over a flapping cycle are cancelled out due to symmetrical flapping wing motion.
Although the relative locations of the peaks of the three forces during downstroke vary a
little bit, the largest force generations are achieved when the wings approach nearly the
middle downstroke. Fig. 17 shows the time courses of aerodynamic torques: aerodynamic
rolling torque (ART), aerodynamic yawing torque (AYT) and aerodynamic pitching
torque (APT). Except for pitching torque, the time-varying rolling torque and yawing
torque produced by left wing are equal to the ones produced by right wings because of
symmetrical flapping wing motion and therefore the overall ART and AYT are zero
during one cycle. Also the overall pitching torque during one flapping cycle is almost
zero. Comparing Fig. 16 with Fig. 17, we can notice that the basic trends of horizontal
force and yawing torque are similar, while the trends of vertical force and rolling torque
are almost the same. The reason is relatively straightforward. As shown from Fig. 11 to
Fig. 13, the forces at the wing tip part are significantly larger than that at the wing root
part, which results in the center of forces far away from the wing root, as is shown in Fig.
18. However, during the middle of downstroke, when the forces are relatively large, the
wings’ position leads to the fact that the arm for the sideslip force Lz is much smaller
than the arm Ly for vertical force.
Fig. 16. Time courses of aerodynamic forces. Left: Vertical (lift) force; Middle:
horizontal (drag or thrust) force; Right: sideslip force.
Fig. 17. Time courses of aerodynamic torques. Left: aerodynamic rolling torque (ART);
Middle: aerodynamic yawing torque (AYT); Right: aerodynamic pitching torque (APT).
Fig. 18. The center of force and arms of
vertical force and sideslip force at the
middle of downstroke.
Fig. 19. Power consumption during one
flapping cycle.
Evaluation of power and aerodynamic efficiency of the hawkmoth in hover
Fig. 19 shows the instantaneous aerodynamic power, which is the power needed to
overcome air resistance. The peak values of power during downstroke and upstroke are
corresponding to that of lift force. The lift to power ratio (power loading) and lift to drag
ratio (L/D ratio) are used to determine the aerodynamic efficiency. The last column in
table 3 shows the average lift, drag, power and the two ratios during one flapping cycle.
Note that the wings produce drag during downstroke and thrust during upstroke. The drag
and thrust cancel each other during one flapping cycle, which result in a very large ratio
of lift to drag. Also we notice that there is a big difference between downstroke and
upstroke.
Downstroke
Upstroke
Down/up
One-cycle
Average lift (N)
2.44E-02
8.70E-03
2.80
1.56E-02
Average drag (N)
1.95E-02
-1.83E-02 1.07
-4.70E-04
Average power (W) 1.03E-01
4.21E-02
2.45
6.79E-02
L/P (N/W)
0.217
0.207
1.05
2.26E-01
Lift/Drag
1.25
0.475
2.63
3.32E+01䦋
Table 3. Average vertical force, horizontal force, power consumption, lift
to power ratio and lift to drag ratio during downstroke, upstroke and one
flapping cycle.
Revolving wing .VS. flapping wing
Lift (N)
Revolving wing 1.70E-02
Flapping wing 1.56E-02
Drag (N) Power (W) L/P (N/W) L/D
1.53E-02 7.10E-02 0.239
1.11
-4.70E-04 6.79E-02 0.226
3.32E+01
Table 4. Comparison of lift, drag, power, lift to power ratio and lift to drag
ratio for revolving wing and flapping wing.
Fig. 20. Vortex structure produced by two revolving wings. The right plot is the vortex
contour on the plane at the 75% of the wing length.
Fig. 21. Time courses of aerodynamic forces (Left) and power consumption (right)
generated by revolving wings and flapping wings respectively.