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First published online February 20, 2004
Journal of Experimental Biology 207, 1137-1150 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.00868
Unsteady aerodynamic forces of a flapping wing
Institute of Fluid Mechanics, Beijing University of Aeronautics & Astronautics, Beijing 100083, People's Republic of China
* Author for correspondence (e-mail: sunmao{at}public.fhnet.cn.net)
Accepted 6 January 2004
The unsteady aerodynamic forces of a model fruit fly wing in flapping
motion were investigated by numerically solving the Navier–Stokes
equations. The flapping motion consisted of translation and rotation [the
translation velocity (ut) varied according to the simple
harmonic function (SHF), and the rotation was confined to a short period
around stroke reversal]. First, it was shown that for a wing of given geometry
with ut varying as the SHF, the aerodynamic force
coefficients depended only on five non-dimensional parameters, i.e. Reynolds
number (Re), stroke amplitude (
), mid-stroke angle of attack
(
m), non-dimensional duration of wing rotation
(
r) and rotation timing [the mean translation velocity
at radius of the second moment of wing area (U), the mean chord
length (c) and c/U were used as reference velocity, length
and time, respectively]. Next, the force coefficients were investigated for a
case in which typical values of these parameters were used (Re=200;
=150°; m=40°; r was 20%
of wingbeat period; rotation was symmetrical). Finally, the effects of varying
these parameters on the force coefficients were investigated.
In the Re range considered (20–1800), when Re was
above 
100, the lift
(
L) and drag
(D) coefficients were
large and varied only slightly with Re (in agreement with results
previously published for revolving wings); the large force coefficients were
mainly due to the delayed stall mechanism. However, when Re was below
100, L decreased and
D increased greatly. At
such low Re, similar to the case of higher Re, the leading
edge vortex existed and attached to the wing in the translatory phase of a
half-stroke; but it was very weak and its vorticity rather diffused, resulting
in the small L and large
D. Comparison of the
calculated results with available hovering flight data in eight species
(Re ranging from 13 to 1500) showed that when Re was above
100, lift equal to insect weight could be produced but when Re
was lower than 100, additional high-lift mechanisms were needed.
In the range of Re above 100, from 90° to 180°
and r from 17% to 32% of the stroke period (symmetrical
rotation), the force coefficients varied only slightly with Re,
and r. This meant that the forces were approximately
proportional to the square of n (n is the wingbeat
frequency); thus, changing and/or n could effectively control
the magnitude of the total aerodynamic force.
The time course of L
(or D) in a half-stroke
for ut varying according to the SHF resembled a half
sine-wave. It was considerably different from that published previously for
ut, varying according to a trapezoidal function (TF) with
large accelerations at stroke reversal, which was characterized by large peaks
at the beginning and near the end of the half-stroke. However, the mean force
coefficients and the mechanical power were not so different between these two
cases (e.g. the mean force coefficients for ut varying as
the TF were approximately 10% smaller than those for ut
varying as the SHF except when wing rotation is delayed).
Key words: flapping wing, insect, computational fluid dynamics, unsteady aerodynamics, delayed stall, force coefficients
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