Quasi-Steady Modeling of an Insect-Like
Flapping Wing
Brett G. Compton
bgcomp2@uky.edu
J. M. McDonough
jmmcd@uky.edu
Department of Mechanical Engineering
University of Kentucky, Lexington, KY 40506
Supported by the Kentucky Space Grant Consortium
APS, Division of Fluid Dynamics 58th Meeting
November 20, 2005, Chicago
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Previous Work
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Munk (NACA Report, 1925)
Weis-Fogh (J. Exp. Biol., 1973)
Maxworthy (J. Fluid Mech., 1979)
Ellington (Phil. Trans. R. Soc. Lond, 1984)
Dickinson (J. Exp. Biol., 1994)
Wang (J. Exp. Biol. 2003)
Sane (J. Exp. Biol., 2003)
Miller and Peskin (J. Exp. Biol., 2004)
Many others
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Physical Problem and Purpose
• Insect flight is both a biological wonder
and an aerodynamic enigma.
• The forces acting on flapping wings are
highly unsteady, leading to higher lift
forces from novel lift-generating
phenomena.
• We want to replicate and explore these
curious aerodynamic phenomena.
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Our Approach
•
We use a quasi-unsteady
approach. Transient
calculations are made at
various virtual wing
orientations throughout the
half-stroke.
– The wing remains fixed and input
conditions are changed to account
for wing rotation.
– A linear velocity profile along the
span is employed to mimic
flapping about the base of the
wing.
– The time steps are chosen such
that between each virtual wing
orientation realistic physical time
passes. No steady-state
calculations are made.
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Our Approach (cont.)
Step 1
Step 2
U
U
Or…
Step 1
Step 2
U
U
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Our Approach (cont.)
• Advantages
– Simplicity
– Accessible Software
– Transient Calculations
at Variable Flight
Regimes
– Laminar/Turbulence
Comparison
• Disadvantages
– Limiting Grid Setup
– Not Quite Realistic
• Rotational Acceleration
not accounted for
• “Ground Effects”
Inherent in Model
• Virtual Mass not
accounted for
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Our Approach (cont.)
Domain seen from the leading
edge
Looking down the span of the wing
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Problem Setup
•
Approximate Bumblebee
Kinematics (Dudley and
Ellington, J. Exp. Biol. 1990a,b)
– R = 1.4 cm
– c = 0.4 cm
– wing thickness = 0.01 cm
– AR = 7
– n = 150 Hz
– Φ = 2 rad (115°)
– αm = 35°
– Ut = 2ΦnR = 8.4 m/s
– Re = cUt/ν = 2333
•
Grid
– Cubic domain 7.5 chord
lengths per side
– ~350,000 grid points
– Wing placed 17 grid points
and ¼ chord lengths from
velocity inlet
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Results – Spiral LEV
Views in each
coordinate direction plus
Pictorial with vortex core.
Physical step 10 of 24.
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Results – Force Coefficients
Average Forces and
Coefficients:
Force Coefficient Time Series
cd-laminar
cd-LES
cl-laminar
cl-LES
1
force coefficient
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
fraction of half-stroke
0.7
0.8
0.9
1
Cl,avg,lam: 0.593
Cl,avg,LES: 0.618
Fl,avg,lam: 0.00296 N
Fl,avg,LES: 0.00309 N
Cd,avg,lam: 0.492
Cd,avg,LES: 0.495
Fd,avg,lam: 0.00246 N
Fd,avg,LES: 0.00248 N
Average Bee weight:
0.001717 N
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Results – Quasi-Steady?
In our model the
Lift and Drag
forces do not
stabilize quickly
enough for the
quasi-steady
model to be
accurate.
Extended Calculations for Coefficient of Lift at Various
Wing Orientations
force coefficients
α = 44.27º
α = 39.19º
α = 35.82º
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
fraction of half-stroke
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Conclusions
• The spiral leading edge vortex was successfully
replicated with this highly simplified model and the
force coefficient time series roughly resemble
results from previous experiments, however…
• A better model is needed. The next step is to use a
research code that will allow more flexibility in the
problem setup, as well as allow imbedded boundary
treatment on the wing.
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