J. Fluid Mech. (2013), vol. 736, R1, doi:10.1017/jfm.2013.555
F lapping dynamics of an inverted flag
Daegyoum Kim†, Julia Cossé, Cecilia Huertas Cerdeira
and Morteza Gharib
Division of Engineering and Applied Science, California Institute of Technology,
Pasadena, CA 91125, USA
(Received 6 September 2013; revised 8 October 2013; accepted 16 October 2013;
first published online 4 November 2013)
The dynamics of an inverted flag are investigated experimentally in order to find
the conditions under which self-excited flapping can occur. In contrast to a typical
flag with a fixed leading edge and a free trailing edge, the inverted flag of our study
has a free leading edge and a fixed trailing edge. The behaviour of the inverted flag
can be classified into three regimes based on its non-dimensional bending stiffness
scaled by flow velocity and flag length. Two quasi-steady regimes, straight mode
and fully deflected mode, are observed, and a limit-cycle flapping mode with large
amplitude appears between the two quasi-steady regimes. Bistable states are found in
both straight to flapping mode transition and flapping to deflected mode transition.
The effect of mass ratio, relative magnitude of flag inertia and fluid inertia, on
the non-dimensional bending stiffness range for flapping is negligible, unlike the
instability of the typical flag. Because of the unsteady fluid force, a flapping sheet can
produce elastic strain energy several times larger than a sheet of the deformed mode,
improving the conversion of fluid kinetic energy to elastic strain energy. According to
the analysis of the leading-edge vortex formation process, the time scale of optimal
vortex formation correlates with efficient conversion to elastic strain energy during
bending.
Key words: aerodynamics, flow–structure interactions
1. Introduction
The interaction of an elastic sheet with a uniform free stream has been studied
to understand flutter of a flag in wind, industrial processes for paper and thin films
(Chang & Moretti 2002; Watanabe et al. 2002), and biological phenomena such as
snoring and animal locomotion (Huang 1995; Zhang et al. 2000; Ristroph & Zhang
2008). In these studies, an elastic sheet had a fixed leading edge and a free trailing
edge, a flag-type configuration. Many studies on the dynamics of the flag have focused
on stability analysis (Guo & Pa
¨
ıdoussis 2000; Tang, Yamamoto & Dowell 2003;
† Email address for correspondence: daegyoum@caltech.edu
c
Cambridge University Press 2013 736 R1-1
D. Kim, J. Cossé, C. Huertas Cerdeira and M. Gharib
x
y
L
x
z
L
A
FIGURE 1. Schematic of an elastic sheet with a free front end and clamped rear end (top and
side views). The dashed straight line is the initial shape of the sheet. A is the distance between
two peaks of the tip in the y-direction, and s is the curvilinear coordinate from the rear end.
Argentina & Mahadevan 2005; Eloy et al. 2008) and the correlation between nonlinear
dynamics and wake structure (Zhang et al. 2000; Connell & Yue 2007; Alben &
Shelley 2008; Michelin, Smith & Glover 2008).
Recently, flow-induced flapping of an elastic sheet has been proposed as a new
approach for harvesting fluid kinetic energy. This method requires fluid kinetic energy
to be converted to strain energy of the structure. Then, the strain energy is converted
to electrical energy with piezoelectric materials (Allen & Smits 2001; Taylor et al.
2001; Akaydin, Elvin & Andreopoulos 2010; Li, Yuan & Lipson 2011; Michelin
& Doar
´
e 2013). However, under the conditions previously studied, the critical flow
velocity required for flapping is high, thereby making this an impractical method of
energy harvesting. Thus, in most experimental studies, an upstream bluff body was
used additionally to induce large oscillation of the downstream structure.
For the successful application of flow-induced oscillation to energy harvesting,
the structure should be designed to easily become unstable at low critical flow
velocity and have high excitation amplitude. In this regard, we study an alternative
configuration to induce self-excited flapping of an elastic sheet. The configuration of
our interest is an inverted flag in which the leading edge is free to move and the
trailing edge is clamped (figure 1); the end conditions are the opposite of a typical
flag. This design was motivated by the fact that a mechanical model with a free front
end and a fixed rear end is generally more susceptible to instability to external axial
loading than its counterpart with reverse end conditions. An inverted pendulum under
gravitational force and a buckling bar under compressive force are such examples. The
vibrating leaves of a tree also give a hint for this inverted flag design (Shao, Chen &
Lin 2012). The leaves can flutter in a breeze regardless of their orientation to wind,
which exemplifies that the flag-type configuration is not a necessary condition for
flow-induced flapping.
To the best of our knowledge, little information is available on the coupling of
an inverted flag configuration with a free-stream flow. Buchak, Eloy & Reis (2010)
conducted an interesting study on the clapping behaviour of a stack of book papers
clamped at a rear end. In their study, weight and elasticity of the papers were
important factors in the oscillation of the papers against a free-stream flow. Guo &
Pa
¨
ıdoussis (2000) studied the linear stability with a potential flow model and claimed
that an inverted elastic sheet was always unstable even in a vanishing free-stream
speed with damping effects neglected, but no further information was provided on
nonlinear dynamics. Here we experimentally investigate the flapping dynamics of an
inverted elastic sheet in order to find how its stability is influenced by parameters
736 R1-2
Flapping dynamics of an inverted flag
such as bending stiffness, flow velocity, fluid density, and sheet length. The effect of
the parameters on drag and elastic strain energy is also examined. In addition, the
flow structure developed by the sheet is identified, and its relationship with flapping
dynamics and strain energy conversion is discussed.
2. Experimental setup
Experiments were conducted in an open-loop wind tunnel composed of an array of
10 × 10 computer fans (Johnson & Jacob 2009). The wind tunnel had a cross-section
of 1.2 m × 1.2 m and was capable of producing free-stream velocity U between 2.2
and 8.5 m s
−1
. The downstream edge of the sheet was clamped vertically between
two long aluminium strips 2.5 cm wide and 1.3 cm thick (figure 1). The sheets were
made from polycarbonate (Young’s modulus E = 2.38 × 10
9
N m
−2
, Poisson’s ratio
ν = 0.38, density ρ
s
= 1.2 × 10
3
kg m
−3
), with thickness h of 0.8 mm. The height
of the sheet H was fixed at 30 cm, and the lengths of the sheet L were 23, 27,
and 30 cm, providing aspect ratios H/L between 1.0 and 1.3. The deformation of
the sheet occurs primarily in the xy-plane, and sagging and twisting of the sheet
due to gravitation were not observed; the deformation was two-dimensional. A small
tip deflection 0.02 < ∆/L < 0.04 was found in the initial sheet configuration due to
material defects.
For the observation of sheet motion, a white plastic tape was attached along the
top edge of the sheet, and it was captured by a high-speed camera (Nanosense
MK3, Dantec Dynamics) mounted over the top of the test section. For each sheet,
images were recorded at 100 frames per second as the wind speed increased from
2.2 to 8.5 m s
−1
. The top edge in the images was detected with a MATLAB script
(Mathworks, Inc.). Aerodynamic drag D acting on the sheet was measured with two
load cells (MB5, Interface, Inc.) connected to the top and bottom of the test section.
The drag was also measured with only the clamping vertical strips and was subtracted
from the total drag in order to obtain the net drag on the sheet.
Two non-dimensional dynamical parameters are important for the study of
interaction between a fluid flow and an elastic sheet. These are non-dimensional
bending stiffness β and mass ratio µ defined as follows (Connell & Yue 2007; Alben
& Shelley 2008; Michelin et al. 2008):
β =
B
ρ
f
U
2
L
3
and µ =
ρ
s
h
ρ
f
L
, (2.1)
where B is the flexural rigidity of the sheet (B = Eh
3
/12(1 − ν
2
)), ρ
f
is fluid density,
and ρ
s
is sheet density. Parameter β characterizes the relative magnitude of the
bending force to the fluid inertial force, and µ describes the relative magnitude of
solid to fluid inertial forces. In the wind tunnel experiments, β ranged from 0.04 to
1.50, and µ ranged from 2.5 to 3.3.
In order to investigate the effect of mass ratio on flapping dynamics, experiments
were also conducted in water for low mass ratio of O(10
−3
). The sheets were clamped
vertically in a free-surface water tunnel with a test section 1.0 m wide and 0.5 m
high. The water velocity ranged between 0.15 and 0.53 m s
−1
. The lengths of the
polycarbonate sheet 0.8 mm thick were 15 cm, 19 cm, and 23 cm. While β ranged
from 0.05 to 1.38, µ was several orders of magnitude lower than that of the wind
tunnel experiments, ranging from 0.004 to 0.006. A camera (IGV-B1920, Imperx, Inc.)
was mounted below the floor of the test section, and the images of the bottom edge of
the sheet were recorded at 10 frames per second as the water speed increases.
736 R1-3
D. Kim, J. Cossé, C. Huertas Cerdeira and M. Gharib
In addition to capturing images of the sheet, planar digital particle image
velocimetry was performed in the water tunnel to visualize vortical structures
of the flapping sheet. The tunnel was seeded with silver-coated hollow ceramic
spheres of diameter 70 µm (AG-SL150-16-TRD, Potters Industries). The particles
were illuminated by an Nd:YAG laser sheet (Gemini PIV, New Wave) at the middle
height of the sheet. Image pairs were captured at a rate of 15 pairs per second,
and processed with PIVview (PIVTEC GmbH). Each pair of images was cross-
correlated with a multi-grid interrogation scheme. The first interrogation window size
was 128×128 pixels with a 50 % overlap, and the final window size was 32×32 pixels
with a 50 % overlap, which produces 119 × 66 grids with the size of 5.8 mm.
3. Results and discussion
3.1. Three dynamical regimes
First, the amplitude and flapping frequency of the elastic sheets are presented for
high mass ratio of O(1). The responses of the sheet can be divided largely into three
modes, depending on non-dimensional bending stiffness β (figure 2). For β higher
than 0.3, the sheet remains straight (straight mode, figure 2c i). For β lower than
0.1, the sheet bends in one direction and maintains a highly curved shape (deflected
mode, figure 2c iv). Even though the sheet flutters slightly in both straight mode and
deflected mode, the peak-to-peak amplitude of the tip A/L is less than 0.2 in these
two modes, and flutter periodicity is not clear. Between these two quasi-static modes
(0.1 < β < 0.3), the sheet flaps from side to side, and the deflection of the sheet is
periodic with nearly constant A/L (limit-cycle flapping mode, figures 2c ii, c iii and
2d). A/L increases drastically in the flapping mode, and plateaus to a range between
1.7 and 1.8. For 0.1 < β < 0.2, the sheet continues to bend past where the tip is
at maximum |y|, which results in slight decrease in the |y|-position of the tip at
maximum deformation as described in figures 2(c iii) and 2(d ). Within the flapping
regime, the Strouhal number f A/U does not show a plateau, unlike A/L, but has a
maximum value of 0.14 around β = 0.2 (figure 2b).
The inverted flag shows several characteristics distinct from those of the typical flag
with a clamped or pinned leading edge and a free trailing edge (e.g. Zhang et al. 2000;
Connell & Yue 2007; Alben & Shelley 2008; Michelin et al. 2008). The inverted flag
exhibits larger peak-to-peak amplitude, up to A/L = 1.7–1.8, than the typical flag. The
large amplitude is realized because the aerodynamic force on the sheet, either lift or
drag, always functions to destabilize the sheet and induce its deformation. In addition,
the critical non-dimensional bending stiffness to initiate large-amplitude flapping from
a straight mode is larger than the critical bending stiffness of the typical flag. The
sheet used in this study was not self-excited when the end conditions of the edges
were reversed to the conventional flag configuration. Moreover, while the typical flag
exhibits periodic or chaotic flapping motion beyond a single critical bending stiffness,
the inverted flag experiences large-amplitude oscillation only within a specific range
of the bending stiffness. This trend of the oscillation within a limited range of the
bending stiffness is somewhat analogous to the amplitude response of an elastically
mounted cylinder in vortex-induced vibration (Shiels, Leonard & Roshko 2001). On
the other hand, an inverted flag with very small bending stiffness beyond the bending
stiffness range studied here may behave differently. A very flexible sheet could bend
180
◦
around the clamped trailing edge and be parallel to the free stream, which results
in the configuration similar to the typical flag. In this case, since the curl at the
736 R1-4
Flapping dynamics of an inverted flag
0 1.0 2.0 3.0 4.0 5.0
–1.0
–0.5
0
0.5
1.0
0 0.1 0.2 0.3 0.4 0.5 0.10 0.2 0.3 0.4 0.5
0.4
0.8
1.2
1.6
2.0
0.04
0.08
0.12
0.16
0.20
(b)(a)
(ci) (cii) (ciii) (civ)
(d)
Flapping modeStraight mode Deflected mode
FIGURE 2. (a) Peak-to-peak amplitude A/L of the tip and (b) Strouhal number f A/U as a
function of bending stiffness β for mass ratio of O(1). H/L = 1.3 (); H/L = 1.1 (
); H/L =
1.0 (4). In (b), only the flapping mode with a constant flapping frequency was considered. In
the straight and deflected modes, the deflection was not periodic. (c) Superimposed sheets at
four bending stiffness values: (i) β = 0.58 (U = 2.8 m s
−1
); (ii) β = 0.26 (U = 4.2 m s
−1
); (iii)
β = 0.10 (U = 6.7 m s
−1
); (iv) β = 0.06 (U = 8.5 m s
−1
). µ = 2.9 and H/L = 1.1. Gravity is
into the paper. See supplementary movies (available at http://dx.doi.org/10.1017/jfm.2013.555)
for (i–iv). (d ) Time history of the y-coordinate of the tip y(s = L)/L; β = 0.26 (dashed) and
β =0.10 (solid). T is a flapping period.
clamped edge does not make much difference compared to the leading edge of the
typical flag, flapping of the typical flag type will be likely to occur.
3.2. Bistability, mode shape, and mass-ratio effect
Subcritical bifurcation and bistable states are found in the inverted flag (figure 3a).
Bistable states exist in both straight–flapping mode transition and flapping–deflected
mode transition. For increasing non-dimensional free-stream velocity U
∗
=
√
1/β =
U
p
ρ
f
L
3
/B, the critical velocities U
∗
c
are 2.1 in the straight–flapping mode bifurcation
and 3.4 in the flapping–deflected mode bifurcation for the sheet of µ = 2.9 and
H/L = 1.1. Then, when U
∗
decreases for the deflected mode, the sheet tends to
maintain its deformed shape and eventually has a slightly lower U
∗
c
(=3.3) than U
∗
c
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