This study experimentally investigates the flow-induced flutter of a thin flexible sheet, focusing on how the sheet's aspect ratio and mass ratio affect its stability and flutter characteristics in the post-critical regime. The flutter frequency of the sheet was obtained using hotwire measurements, while flutter amplitude and mode shape were acquired through high-speed imaging. The flowfield around the flapping sheet was analyzed using particle image velocimetry (PIV). Based on experimental observations, we report the onset of flutter as a subcritical bifurcation with hysteresis. The dynamic characteristics of the sheet play a significant role in its flutter instability, with the onset and cessation of flutter occurring at a frequency close to the sheet's second-mode natural frequency. The results show that both aspect ratio and mass ratio significantly affect the critical wind speed and flutter characteristics in the post-critical regime. Both flutter frequency and amplitude decrease as the aspect ratio decreases. PIV measurements in various planes reveal the highly three-dimensional nature of the flow. Results from off-axis PIV show a pair of counter-rotating spiral vortices in the wake that oscillate and change orientation with the sheet's movement. Additionally, a theoretical analysis was conducted to derive an approximate analytical relationship between the aspect ratio and critical wind speed. Experimental results aligned well with theoretical predictions for sheets with low aspect ratios (aspect ratio 1) but deviated for sheets with higher aspect ratios (aspect ratio >1). The relevant scaling parameters have also been explored to represent the experimental data in a non-dimensional form.

1.
Adrian
,
R. J.
and
Westerweel
,
J.
,
Particle Image Velocimetry
. Number 30. (
Cambridge University Press
,
2011
).
2.
Akcabay
,
D. T.
and
Young
,
Y. L.
, “
Hydroelastic response and energy harvesting potential of flexible piezoelectric beams in viscous flow
,”
Phys. Fluids
24
(
5
),
054106
(
2012
).
3.
Alben
,
S.
, “
Flag flutter in inviscid channel flow
,”
Phys. Fluids
27
(
3
),
033603
(
2015
).
4.
Alben
,
S.
and
Shelley
,
M. J.
, “
Flapping states of a flag in an inviscid fluid: Bistability and the transition to chaos
,”
Phys. Rev. Lett.
100
(
7
),
074301
(
2008
).
5.
Chen
,
M.
,
Jia
,
L.-B.
,
Wu
,
Y.-F.
,
Yin
,
X.-Z.
, and
Ma
,
Y.-B.
, “
Bifurcation and chaos of a flag in an inviscid flow
,”
J. Fluids Struct.
45
,
124
137
(
2014
).
6.
Coleman
,
H. W.
and
Steele
,
W. G.
,
Experimentation, Validation, and Uncertainty Analysis for Engineers
(
John Wiley and Sons
,
2018
).
7.
Doaré
,
O.
,
Sauzade
,
M.
, and
Eloy
,
C.
, “
Flutter of an elastic plate in a channel flow: Confinement and finite-size effects
,”
J. Fluids Struct.
27
,
76
88
(
2011
).
8.
Dunnmon
,
J. A.
,
Stanton
,
S. C.
,
Mann
,
B. P.
, and
Dowell
,
E. H.
, “
Power extraction from aeroelastic limit cycle oscillations
,”
J. Fluids Struct.
27
,
1182
1198
(
2011
).
9.
Eloy
,
C.
,
Kofman
,
N.
, and
Schouveiler
,
L.
, “
The origin of hysteresis in the flag instability
,”
J. Fluid Mech.
691
,
583
593
(
2012
).
10.
Eloy
,
C.
,
Lagrange
,
R.
,
Souilliez
,
C.
, and
Schouveiler
,
L.
, “
Aeroelastic instability of cantilevered flexible plates in uniform flow
,”
J. Fluid Mech.
611
,
97
106
(
2008
).
11.
Eloy
,
C.
,
Souilliez
,
C.
, and
Schouveiler
,
L.
, “
Flutter of a rectangular plate
,”
J. Fluids Struct.
23
(
6
),
904
919
(
2007
).
12.
Gibbs
,
S. C.
,
Wang
,
I.
, and
Dowell
,
E.
, “
Theory and experiment for flutter of a rectangular plate with a fixed leading edge in three-dimensional axial flow
,”
J. Fluids Struct.
34
,
68
83
(
2012
).
13.
Hellum
,
A.
,
Mukherjee
,
R.
, and
Hull
,
A. J.
, “
Flutter instability of a fluid-conveying fluid-immersed pipe affixed to a rigid body
,”
J. Fluids Struct.
27
(
7
),
1086
1096
(
2011
).
14.
Hua
,
R.-N.
,
Zhu
,
L.
, and
Lu
,
X.-Y.
, “
Locomotion of a flapping flexible plate
,”
Phys. Fluids
25
(
12
),
121901
(
2013
).
15.
Huang
,
L.
, “
Flutter of Cantilevered Plates in Axial Flow
,”
J. Fluids Struct.
9
,
127
147
(
1995
).
16.
Huber
,
G.
, “
Swimming in Flat sea
,”
Nature
408
,
777
778
(
2000
).
17.
Jin
,
Y.
and
Chamorro
,
L.
, “
Passive pitching of splitters in the trailing edge of elliptic cylinders
,”
J. Fluid Mech.
826
,
363
375
(
2017
).
18.
Jin
,
Y.
,
Cheng
,
S.
, and
Chamorro
,
L. P.
, “
Active pitching of short splitters past a cylinder: Drag increase and wake
,”
Phys. Rev. E
100
(
6
),
063106
(
2019
).
19.
Kumar
,
D.
,
Arekar
,
A. N.
, and
Poddar
,
K.
, “
The dynamics of flow-induced flutter of a thin flexible sheet
,”
Phys. Fluids
33
(
3
),
034131
(
2021
).
20.
Kumar
,
S. K.
,
Bose
,
C.
,
Ali
,
S. F.
,
Sarkar
,
S.
, and
Gupta
,
S.
, “
Investigations on a vortex induced vibration based energy harvester
,”
Appl. Phys. Lett.
111
(
24
),
243903
(
2017
).
21.
Liao
,
J. C.
, “
A review of fish swimming mechanics and behaviour in altered flows
,”
Phil. Trans. R Soc. B
362
(
1487
),
1973
1993
(
2007
).
22.
Pang
,
Z.
,
Jia
,
L-b.
, and
Yin
,
X-z.
, “
Flutter instability of rectangle and trapezoid flags in uniform flow
,”
Phys. Fluids
22
(
12
),
121701
(
2010
).
23.
Raffel
,
M.
,
Willert
,
C. E.
,
Scarano
,
F.
,
Kähler
,
C. J.
,
Wereley
,
S. T.
, and
Kompenhans
,
J.
,
Particle Image Velocimetry: A Practical Guide
(
Springer
,
2018
).
24.
Rips
,
A.
and
Mittal
,
R.
, “
Flutter-enhanced mixing in small-scale mixers
,”
Phys. Fluids
31
(
10
),
107107
(
2019
).
25.
Rips
,
A.
,
Shoele
,
K.
, and
Mittal
,
R.
, “
Heat transfer enhancement in laminar flow heat exchangers due to flapping flags
,”
Phys. Fluids
32
(
6
),
063603
(
2020
).
26.
Ryu
,
J.
,
Yang
,
J.
,
Park
,
S. G.
, and
Sung
,
H. J.
, “
Phase-mediated locomotion of two self-propelled flexible plates in a tandem arrangement
,”
Phys. Fluids
32
(
4
),
041901
(
2020
).
27.
Sharma
,
K.
and
Dutta
,
S.
, “
Flow control over a square cylinder using attached rigid and flexible splitter plate at intermediate flow regime
,”
Phys. Fluids
32
(
1
),
014104
(
2020
).
28.
Sharma
,
K. R.
and
Dutta
,
S.
, “
Influence of length and effective stiffness of an attached flexible foil for flow over a square cylinder
,”
J. Fluids Struct.
104
,
103298
(
2021
).
29.
Sharma
,
V.
and
Dutta
,
S.
, “
Effect on drag–thrust transition for flapping airfoil with chordwise flexibility
,”
Phys. Fluids
35
(
7
),
074103
(
2023
).
30.
Shelley
,
M. J.
and
Zhang
,
J.
, “
Flapping and Bending Bodies Interacting with Fluid Flows
,”
Annu. Rev. Fluid Mech.
43
,
449
465
(
2011
).
31.
Shen
,
L.
,
Zhang
,
X.
,
Yue
,
D. K.
, and
Triantafyllou
,
M. S.
, “
Turbulent flow over a flexible wall undergoing a streamwise traveling wavy motion
,”
J. Fluid Mech.
484
,
197
221
(
2003
).
32.
Shi
,
G.
,
Xiao
,
Q.
, and
Zhu
,
Q.
, “
Effects of time-varying flexibility on the propulsion performance of a flapping foil
,”
Phys. Fluids
32
(
12
),
121904
(
2020
).
33.
Shinde
,
S. Y.
and
Arakeri
,
J. H.
, “
Jet meandering by a foil pitching in quiescent fluid
,”
Phys. Fluids
25
(
4
),
041701
(
2013
).
34.
Shinde
,
S. Y.
and
Arakeri
,
J. H.
, “
Flexibility in flapping foil suppresses meandering of induced jet in absence of free stream
,”
J. Fluid Mech.
757
,
231
250
(
2014
).
35.
Shoele
,
K.
and
Mittal
,
R.
, “
Computational study of flow-induced vibration of a reed in a channel and effect on convective heat transfer
,”
Phys. Fluids
26
(
12
),
127103
(
2014
).
36.
Shyy
,
W.
,
Berg
,
M.
, and
Ljungqvist
,
D.
, “
Flapping and flexible wings for biological and micro air vehicles
,”
Prog. Aerosp. Sci.
35
(
5
),
455
505
(
1999
).
37.
Stamhuis
,
E.
and
Thielicke
,
W.
, “
Pivlab–towards user-friendly, affordable and accurate digital particle image velocimetry in matlab
,”
J. open research software
2
(
1
),
30
(
2014
).
38.
Sunil
,
P.
,
Kumar
,
S.
, and
Poddar
,
K.
, “
Flow past a rotationally oscillating cylinder with an attached flexible filament
,”
J. Fluid Mech.
930
,
A3
(
2022
).
39.
Sunil
,
P.
,
Kumar
,
S.
, and
Poddar
,
K.
, “
Effect of filament length on the flow field of a rotationally oscillating cylinder with an attached filament
,”
Phys. Fluids
35
(
12
),
123605
(
2023
).
40.
Taneda
,
S.
, “
Waving motions of flags
,”
J. Phys. Soc. Jpn.
24
,
392
401
(
1968
).
41.
Tang
,
D.
,
Levin
,
D.
, and
Dowell
,
E.
, “
Experimental and theoretical correlations for energy harvesting from a large flapping flag response
,”
J. Fluids Struct.
86
,
290
315
(
2019
).
42.
Tang
,
D. M.
,
Yamamoto
,
H.
, and
Dowell
,
E. H.
, “
Flutter and limit cycle oscillations of two-dimensional panels in three-dimensional axial flow
,”
J. Fluids Struct.
17
,
225
242
(
2003
).
43.
Tang
,
L.
and
Païdoussis
,
M. P.
, “
The influence of the wake on the stability of cantilevered flexible plates in axial flow
,”
J. Sound Vib.
310
(
3
),
512
526
(
2008
).
44.
Tang
,
L.
and
Païdoussis
,
M. P
, “
On the instability and the post-critical behaviour of two-dimensional cantilevered flexible plates in axial flow
,”
J. Sound Vib.
305
(
1–2
),
97
115
(
2007
).
45.
Thielicke
,
W.
, “
The flapping flight of birds: Analysis and application
,”
Ph. D. thesis
(
University of Groningen
,
2014
).
46.
Triantafyllou
,
M. S.
and
Triantafyllou
,
G. S.
, “
An efficient swimming machine
,”
Sci. Am.
272
(
3
),
64
70
(
1995
).
47.
Vineeth
,
V.
,
Patel
,
D. K.
,
Roy
,
S.
,
Goli
,
S.
, and
Roy
,
A.
, “
Investigations into transient wakes behind a custom airfoil undergoing pitching motion
,”
Eur. J. Mech.-B/Fluids
85
,
193
213
(
2021
).
48.
Wang
,
X.
,
Alben
,
S.
,
Li
,
C.
, and
Young
,
Y. L.
, “
Stability and scalability of piezoelectric flags
,”
Phys. Fluids
28
(
2
),
023601
(
2016
).
49.
Watanabe
,
Y.
,
Isogai
,
K.
,
Suzuki
,
S.
, and
Sugihara
,
M.
, “
A theoretical study of paper flutter
,”
J. Fluids Struct.
16
(
4
),
543
560
(
2002a
).
50.
Watanabe
,
Y.
,
Suzuki
,
S.
,
Sugihara
,
M.
, and
Sueoka
,
Y.
, “
An experimental study of paper flutter
,”
J. Fluids Struct.
16
(
4
),
529
542
(
2002b
).
51.
Wilson
,
B. M.
and
Smith
,
B. L.
, “
Uncertainty on piv mean and fluctuating velocity due to bias and random errors
,”
Meas. Sci. Technol.
24
(
3
),
035302
(
2013
).
52.
Yamaguchi
,
N.
,
Sekiguchi
,
T.
,
Yokota
,
K.
, and
Tsujimoto
,
Y.
, “
Flutter limits and behavior of a flexible thin sheet in high-speed flow - II: Experimental results and predicted behaviors for low mass ratios
,”
J. Fluids Eng
122
(
1
),
74
83
(
2000
).
53.
Yu
,
Y.
,
Liu
,
Y.
, and
Amandolese
,
X.
, “
A Review on Fluid-Induced Flag Vibrations
,”
Appl. Mech. Rev.
71
(
1
),
010801
(
2019
).
54.
Yu
,
Y.
,
Liu
,
Y.
, and
Chen
,
Y.
, “
Vortex dynamics and heat transfer behind self-oscillating inverted flags of various lengths in channel flow
,”
Phys. Fluids
30
(
4
),
045104
(
2018
).
55.
Zhao
,
W.
,
Païdoussis
,
M. P.
,
Tang
,
L.
,
Liu
,
M.
, and
Jiang
,
J.
, “
Theoretical and experimental investigations of the dynamics of cantilevered flexible plates subjected to axial flow
,”
J. Sound Vib.
331
,
575
587
(
2012
).
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