Under consideration for publication in J. Fluid Mech.
1
Global modes and nonlinear analysis of
inverted-flag flapping
Andres Goza
1
†
, Tim Colonius
1
, and John E. Sader
2,3
1
Department of Mechanical and Civil Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
2
ARC Centre of Excellence in Exciton Science, School of Mathematics and Statistics,
University of Melbourne, Victoria 3010, Australia
3
Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
(Received xx; revised xx; accepted xx)
An inverted flag has its trailing edge clamped and exhibits dynamics distinct from that
of a conventional flag, whose leading edge is restrained. We perform nonlinear simulations
and a global stability analysis of the inverted-flag system for a range of Reynolds numbers,
flag masses and stiffnesses. Our global stability analysis is based on a linearisation of the
fully-coupled fluid-structure system of equations. The calculated equilibria are steady-
state solutions of the fully-coupled nonlinear equations. By implementing this approach,
we (i) explore the mechanisms that initiate flapping, (ii) study the role of vortex shedding
and vortex-induced vibration (VIV) in large-amplitude flapping, and (iii) characterise the
chaotic flapping regime. For point (i), we identify a deformed-equilibrium state and show
through a global stability analysis that the onset of flapping is due to a supercritical
Hopf bifurcation. For large-amplitude flapping, point (ii), we confirm the arguments of
Sader
et al.
(2016
a
) that for a range of parameters this regime is a VIV. We also show
that there are other flow regimes for which large-amplitude flapping persists and is not
a VIV. Specifically, flapping can occur at low Reynolds numbers (
<
50), albeit via a
previously unexplored mechanism. Finally, with respect to point (iii), chaotic flapping
has been observed experimentally for Reynolds numbers of
O
(10
4
), and here we show that
chaos also persists at a moderate Reynolds number of 200. We characterise this chaotic
regime and calculate its strange attractor, whose structure is controlled by the above-
mentioned deformed equilibria and is similar to a Lorenz attractor. These results are
contextualised with bifurcation diagrams that depict the different equilibria and various
flapping regimes.
1. Introduction
Uniform flow past a conventional flag—where the flag is pinned or clamped at its
leading edge with respect to the oncoming flow—has been studied widely beginning with
the early work of Taneda (1968) (see Shelley & Zhang (2011) for a recent review). By
contrast, studies of flow past an inverted flag, in which the flag is clamped at its trailing
edge, have only been reported recently. The inverted-flag system displays a wide range of
dynamical regimes (Kim
et al.
2013; Gurugubelli & Jaiman 2015; Ryu
et al.
2015), many
of which are depicted in figure 1. This figure is produced from the numerical simulations
described in section 2.
One of the dynamical regimes depicted is large-amplitude flapping (figure 1d), which
is associated with a larger strain energy than that of conventional flag flapping. These
†
Email address for correspondence: ajgoza@gmail.com
arXiv:1709.09745v1 [physics.flu-dyn] 27 Sep 2017
2
A. Goza, T. Colonius, and J. Sader
(a)
(b)
(c)
(d)
(e)
Figure 1: Time lapses of flag position for (a) the undeformed equilibrium, (b) small-
deflection stable, (c) small-deflection deformed flapping, (d) large-amplitude flapping,
and (e) deflected-mode regimes. In all figures, the flag is clamped at its right edge and
the flow direction is from left to right.
large bending strains make the inverted-flag system a promising candidate for energy har-
vesting technologies that convert strain energy to electricity,
e.g.
, by using piezoelectric
materials. Shoele & Mittal (2016) studied this energy harvesting potential in detail by
performing numerical simulations of a fully-coupled fluid-structure-piezoelectric model.
Transitions between the various regimes in figure 1 depend on the Reynolds number
(
Re
), dimensionless mass ratio (
M
ρ
), and dimensionless bending stiffness (
K
B
), defined
as
Re
=
ρ
f
UL
μ
, M
ρ
=
ρ
s
h
ρ
f
L
, K
B
=
EI
ρ
f
U
2
L
3
(1.1)
where
ρ
f
(
ρ
s
) is the fluid (structure) density,
U
is the freestream velocity,
L
is the flag
length,
μ
is the shear viscosity of the fluid,
h
is the flag thickness, and
EI
is the flexural
rigidity of the flag. In experiments, regime transitions are triggered by increasing the
flow rate (Kim
et al.
2013). This coincides with a decrease in
K
B
and an increase in
Re
for fixed
M
ρ
, by virtue of (1.1). In contrast, numerical simulations often decrease the
flag’s stiffness at fixed
Re
and
M
ρ
, which isolates the effect of various parameters and
facilitates comparison to previous numerical studies of flow-induced vibration.
Simulations show that for moderate Reynolds numbers (
.
1000), a systematic decrease
in
K
B
causes a change from a stable undeformed equilibrium state (figure 1a) to a small-
deflection stable state (figure 1b). This is followed by a transition to small-deflection
deformed flapping (figure 1c), then to large-amplitude flapping (figure 1d), and finally
to a deflected-mode regime (figure 1e) (Gurugubelli & Jaiman 2015; Ryu
et al.
2015).
These simulations have been performed primarily for
M
ρ
6
O
(1) (heavy fluid loading),
though Shoele & Mittal (2016) considered large values of
M
ρ
.
The same regime transitions persist at higher Reynolds numbers,
Re
∼
O
(10
4
), except
that the small-deflection stable and small-deflection deformed flapping regimes discussed
above are no longer present. That is, the undeformed equilibrium directly gives way to
large-amplitude flapping (Kim
et al.
2013). Moreover, Sader
et al.
(2016
a
) experimentally
identified a chaotic flapping regime (not shown in figure 1) at these higher Reynolds
numbers that has yet to be reported using numerical simulations with
Re
6
O
(1000).
At low Reynolds numbers (
Re <
50), numerical simulations have shown that the
inverted flag’s dynamics can change significantly: no flapping occurs, with the only
observed regimes being the undeformed equilibrium and stable deflected states (Ryu
et al.
2015). These simulations were performed over a wide range of
K
B
for only one
value of
M
ρ
, and the system’s dependence on these two parameters remains an open
question at these lower Reynolds numbers.
Several driving mechanisms of the various regimes illustrated in figures 1(a)–(e) have
been identified. The bifurcation from the undeformed equilibrium is caused by a di-
Global modes and nonlinear analysis of inverted-flag flapping
3
vergence instability (
i.e.
, the instability is independent of
M
ρ
). This mechanism was
originally suggested by Kim
et al.
(2013), and subsequently found computationally
(Gurugubelli & Jaiman 2015) and mathematically via a linear stability analysis (Sader
et al.
2016
a
). For large-amplitude flapping, Sader
et al.
(2016
a
) used experiments and
a scaling analysis to argue that this regime is a vortex-induced vibration (VIV) for a
distinct range of parameters. The primary role of vortex shedding in large-amplitude
flapping is further evidenced by the above-mentioned observation of Ryu
et al.
(2015)
that flapping does not occur below
Re
≈
50 (for certain values of
M
ρ
). Based on a scaling
analysis, Sader
et al.
(2016
a
) also predicted that VIV should cease as the mass ratio,
M
ρ
, increases—a prediction that is yet to be verified. With respect to the deflected-
mode regime, small-amplitude flapping about a large mean-deflected position occurs,
and Shoele & Mittal (2016) showed that the flapping frequency is identical to that of the
vortex shedding caused by the flag’s bluffness.
In this article, we use high-fidelity nonlinear simulations and a global linear stability
analysis to further characterise the regimes in figure 1 and explore their driving physical
mechanisms. We emphasise that our global stability analysis is based on a linearisation of
the fully-coupled fluid-structure system of equations. Moreover, the computed equilibria
are steady-state solutions of the fully-coupled nonlinear equations described in section
2. Our results are presented for Reynolds numbers of 20 and 200, various values of
K
B
,
and values of
M
ρ
spanning four orders of magnitude.
Using this approach, we (i) study the mechanisms responsible for the onset of small-
deflection deformed flapping, (ii) probe the role of vortex shedding and VIV in large-
amplitude flapping, and (iii) investigate whether chaotic flapping occurs at low-to-
moderate Reynolds numbers (
Re
= 20 and 200). To explore (i), we first demonstrate
that the small-deflection stable state is an equilibrium of the fully-coupled fluid-structure
system. Through a global stability analysis, we show that the subsequent transition
to small-deflection deformed flapping (figure 1c) as the bending stiffness decreases is a
supercritical Hopf bifurcation of this deformed equilibrium. For point (ii), we confirm the
arguments of Sader
et al.
(2016
a
) that large-amplitude flapping is a VIV for the higher
Reynolds number of
Re
= 200 and lower values of the mass ratio,
M
ρ
< O
(1). VIV is also
shown to cease for sufficiently large
M
ρ
, consistent with the scaling analysis of Sader
et al.
(2016
a
). Moreover, we show that large-amplitude flapping persists at these large values
of
M
ρ
despite the absence of VIV, albeit by a previously unidentified mechanism. This
non-VIV large-amplitude flapping regime also occurs for large
M
ρ
at the lower Reynolds
number of
Re
= 20. Consistent with the simulation results of Ryu
et al.
(2015), we find
no flapping at this low Reynolds number for
M
ρ
< O
(1). Finally, with respect to (iii), we
confirm that chaotic flapping persists at moderate Reynolds numbers (
Re
= 200) for light
flags with
M
ρ
<
1, and demonstrate that the structure of the associated strange attractor
is controlled by a combination of the large-amplitude and deflected-mode regimes. Chaos
does not occur for heavy flags at
Re
= 200 or for any mass ratio considered at
Re
= 20.
Thus, chaos is associated with parameters for which VIV flapping occurs.
We contextualise the simulation results over this wide range of parameters using
bifurcation diagrams. These provide an overview of the equilibria, their stability, and the
flapping dynamics. Figure 2 shows an
illustrative
bifurcation diagram that summarises
what will be shown in later sections. In these bifurcation diagrams, the leading edge
transverse displacement (tip displacement) is plotted versus the flag flexibility (1
/K
B
)
for a particular choice of
Re
and
M
ρ
(see the caption for details). Note that even though
the undeformed and deformed equilibria become unstable with a decrease in
K
B
, they
nonetheless remain as equilibria of the system. We demonstrate below through a global
4
A. Goza, T. Colonius, and J. Sader
Figure 2: A schematic bifurcation diagram that summarises the results obtained for
various parameters considered in the present work. Equilibria are presented by lines (
,
stable equilibria;
, unstable equilibria;
, stability depends on parameters). The
lines with the double arrows indicate regimes where flapping occurs, with the top and
bottom lines representing the peak-to-peak flapping amplitude. The diagram shows that
with decreasing
K
B
(moving left to right), the system transitions from the undeformed
equilibrium to a stable deformed equilibrium. Following this, the system bifurcates to
small-deflection deformed flapping, then large-amplitude flapping (chaotic flapping can
also occur in this regime depending on parameters), and finally to a deflected-mode
regime whose dynamics depend on Reynolds number: for
Re
= 20, no flapping occurs
and the large-deflection state is an equilibrium of the fully-coupled system; for
Re
= 200,
the large-deflection state is characterised by small-amplitude flapping. The above diagram
only corresponds to cases when flapping occurs. For
Re
= 20 and
M
ρ
< O
(1), no flapping
occurs and the only two regimes are the undeformed and deformed equilibria.
stability analysis that these unstable deformed equilibria are key to understanding a
variety of flapping behaviour of the inverted-flag system.
Two-dimensional (2D) simulations are presented throughout. As mentioned above,
many similarities exist between the 3D experiments of Kim
et al.
(2013) and 2D simula-
tions of Gurugubelli & Jaiman (2015); Ryu
et al.
(2015); Shoele & Mittal (2016). This
suggests that features of the 2D dynamics persist in 3D (though Sader
et al.
(2016
b
)
demonstrated that substantial differences occur for low-aspect ratio flags). Exploring
these similarities and differences between 2D and 3D geometries is a subject of future
work and is not considered here. Quantities presented below are dimensionless, with
length scales, velocity scales, and time scales nondimensionalised by
L
,
U
, and
L/U
,
respectively.
Global modes and nonlinear analysis of inverted-flag flapping
5
2. Numerical methods: nonlinear solver and global stability analysis
Our nonlinear simulations use the immersed boundary method of Goza & Colonius
(2017). The method treats the fluid with the 2D Navier-Stokes equations, and the flag
with the geometrically nonlinear Euler-Bernoulli beam equation. The method is strongly-
coupled (
i.e.
, it accounts for the nonlinear coupling between the flag and the fluid), and
therefore allows for arbitrarily large flag displacements and rotations. We have validated
our method against a variety of test problems involving conventional and inverted
flags (Goza & Colonius 2017). The global stability analysis is based on a linearisation
of the nonlinear, fully-coupled flow-structure interaction system, and therefore reveals
instability-driving mechanisms in both the flag and the fluid.
In what follows, we review the nonlinear solver (see Goza & Colonius (2017) for more
details) and derive the linearised equations. We then describe the global mode solution
approach, the procedure used to compute equilibria of the flow-flag system, and the grid
spacing and domain size used for our simulations.
2.1.
Nonlinear solver
We define the fluid domain as
Ω
and the flag surface as
Γ
. We let
x
denote the Eulerian
coordinate representing a position in space, and
χ
(
θ,t
) be the Lagrangian coordinate
attached to the body
Γ
(
θ
is a variable that parametrizes the surface). The dimensionless
governing equations are written as
∂
u
∂t
=
−
u
·∇
u
−∇
p
+
1
Re
∇
2
u
+
∫
Γ
f
(
χ
(
θ,t
))
δ
(
χ
(
θ,t
)
−
x
)
dθ
(2.1)
∇·
u
= 0
(2.2)
ρ
s
ρ
f
∂
2
χ
∂t
2
=
1
ρ
f
U
2
∇·
σ
+
g
(
χ
)
−
f
(
χ
)
(2.3)
∫
Ω
u
(
x
)
δ
(
x
−
χ
(
θ,t
))
d
x
=
∂
χ
(
θ,t
)
∂t
(2.4)
In the above, (2.1) expresses the Navier-Stokes equations in an immersed boundary
formulation, (2.2) is the continuity equation for the fluid, (2.3) represents the structural
equations governing the motion of the flag (
g
is a body force term), and (2.4) is the
no-slip boundary condition enforcing that the fluid velocity matches the flag velocity on
the flag surface. Note that
f
represents the effect from the flag surface stresses on the
fluid, and is present in both (2.1) and (2.3) since by Newton’s third law its negative
imparts the fluid stresses on the flag surface (Goza & Colonius 2017). In (2.3), the time
derivative is a Lagrangian derivative and the stress tensor is the Cauchy tensor in terms
of the deformed flag configuration.
The fluid equations are spatially discretised with the immersed boundary discrete-
streamfunction formulation of Colonius & Taira (2008), which removes the pressure and
eliminates the continuity equation. The flag equations are treated with a finite element
corotational formulation (Criesfield 1991). The spatially discrete, temporally continuous
equations written as a first order system of differential-algebraic equations are
C
T
C
̇
s
=
−
C
T
N
(
s
) +
1
Re
C
T
LCs
−
C
T
E
T
(
χ
)
f
(2.5)
M
̇
ζ
=
−
R
(
χ
) +
Q
(
g
+
W
(
χ
)
f
)
(2.6)
̇
χ
=
ζ
(2.7)
0 =
E
(
χ
)
Cs
−
ζ
(2.8)
6
A. Goza, T. Colonius, and J. Sader
where
χ
and
f
are discrete analogues to their continuous counterparts,
s
is the discrete
streamfunction,
ζ
is the flag velocity, and all other variables are defined below.
Equation (2.5) represents the Navier-Stokes equations written in a discrete-
streamfunction formulation, (2.6) is the geometrically nonlinear Euler-Bernoulli beam
equation, (2.7) matches the time derivative of the flag position to the flag velocity, and
(2.8) is the interface constraint that the fluid and flag must satisfy the no-slip boundary
condition on the flag surface.
In (2.5)–(2.8),
C
and
C
T
are discrete curl operators that mimic
∇ ×
(
·
);
N
(
s
) is
a discretization of the advection operator
u
· ∇
u
written in terms of the discrete
streamfunction (Colonius & Taira 2008);
L
is a discrete Laplacian associated with
the viscous diffusion term;
E
T
f
is a “smearing” operator (arising from the immersed
boundary treatment) that applies the surface stresses from the flag onto the fluid;
M
is
a mass matrix associated with the flag’s inertia;
R
(
χ
) is the internal stress within the
flag;
Qg
is a body force term (
e.g.
, gravity); and
QWf
is the stress imposed on the flag
from the fluid.
Equation (2.5) is discretised in time using an Adams Bashforth AB2 scheme for the
convective term and a second order Crank-Nicholson scheme for the diffusive term. The
flag equations (2.6)–(2.7) are discretized using an implicit Newmark scheme. The method
is strongly coupled, so the constraint equation (2.8) is enforced at each time step including
the present one.
A novel feature of our method is the efficient iterative procedure used to treat the
nonlinear coupling between the flag and fluid. Many methods use a block-Gauss Seidel
iterative procedure, which converges slowly (or not at all) for light structures (Tian
et al.
2014). Other methods use a Newton-Raphson scheme, which exhibits fast convergence
behaviour but requires the solution of linear systems involving large Jacobian matrices
(Degroote
et al.
2009). Our method employs the latter approach, but we use a block-
LU factorization of the Jacobian matrix to restrict all iterations to subsystems whose
dimensions scale with the number of discretisation points on the flag, rather than on
the entire flow domain. Thus, our algorithm inherits the fast convergence behaviour
of Newton-Raphson methods while substantially reducing the cost of performing an
iteration.
2.2.
Linearised equations and global modes
For ease of notation, we define the state vector
y
= [
s,ζ,χ,f
]
T
and let
r
(
y
) be the right
hand side of (2.5)–(2.8). We write the state as
y
=
y
b
+
y
p
, where
y
b
= [
s
b
,ζ
b
,χ
b
,f
b
]
T
is a base state and
y
p
= [
s
p
,ζ
p
,χ
p
,f
p
]
T
is a perturbation. Plugging this expression for
y
into (2.5)–(2.8), Taylor expanding about
y
b
, and retaining only first order terms in the
perturbation variables gives the linearised equations:
B
̇
y
p
=
A
(
y
b
)
y
p
(2.9)
where
B
=
C
T
C
M
I
0
, A
(
y
b
) =
J
ss
0
−
J
χs
−
C
T
E
T
0
0
−
K
+
J
χχ
QW
0
I
0
0
EC
−
I
J
χc
0
y
=
y
b
(2.10)
Global modes and nonlinear analysis of inverted-flag flapping
7
and the remaining sub-blocks of the Jacobian matrix
A
are given in index notation as
(
J
ss
)
ik
=
−
(
C
T
C
)
2
ik
−
C
T
ij
∂
N
j
∂s
k
(2.11)
(
J
χs
)
ik
=
C
T
ij
∂E
T
jl
∂χ
k
(
f
b
)
l
(2.12)
(
J
χχ
)
ik
=
Q
ij
∂W
jl
∂χ
k
(
f
b
)
l
(2.13)
(
J
χc
)
ik
=
∂E
ij
∂χ
k
C
jl
(
s
b
)
l
(2.14)
Note that we used
B
̇
y
b
=
r
(
y
b
) in arriving at the linearised equations (2.9).
Global modes are eigenvectors
v
of the generalised eigenvalue problem
Av
=
λBv
,
where
λ
is the corresponding eigenvalue. We build and store
A
and
B
sparsely and solve
the generalised eigenvalue problem using an implicitly restarted Arnoldi algorithm (see
Lehoucq
et al.
(1998) for more details).
In the results below, 1
×
10
−
10
was used as the tolerance for convergence of the
computed eigenvalues and eigenvectors. Global eigenfunctions are unique to a scalar
multiple, and were scaled to unit norm,
||
y
||
2
= 1.
2.3.
Equilibrium computations
Undeformed and deformed equilibria are steady state solutions to the fully-coupled
equations (2.5)-(2.8) with all time derivate terms set to zero;
i.e.
, these equilibria satisfy
0 =
r
(
y
), where
y
= [
s,ζ,χ,f
]
T
is the state vector and
r
(
y
) is the right hand side
of (2.5)-(2.8). This is a nonlinear algebraic system of equations that we solve using a
Newton-Raphson method. With this method, the
k
th
guess for the base state,
y
(
k
)
, is
updated as
y
(
k
+1)
=
y
(
k
)
+
∆y
, where
∆y
=
−
(
A
(
y
(
k
)
))
−
1
r
(
y
(
k
)
)
(2.15)
Note that the Jacobian matrix
A
in (2.15) is the same matrix as in (2.10) evaluated at
y
=
y
(
k
)
.
The guess for the state
y
is updated until the residual at the current guess is less than
a desired threshold (
i.e.
, until
||
r
(
y
(
k
))
||
2
/
||
y
(
k
)
||
2
<
). In the results shown below we
used
= 1
×
10
−
6
.
2.4.
Domain size and grid resolution
The flow equations are treated using a multidomain approach: the finest grid surrounds
the body and grids of increasing coarseness are used at progressively larger distances
(Colonius & Taira 2008). In all computations below, the domain size of the finest
sub-domain is [
−
0
.
2
,
1
.
8]
×
[
−
1
.
1
,
1
.
1] and the total domain size is [
−
15
.
04
,
16
.
64]
×
[
−
17
.
44
,
17
.
44]. The grid spacing on the finest domain is
h
= 0
.
01 and the grid spacing
for the flag is
∆s
= 0
.
02. For computations involving time marching, the time step is
∆t
= 0
.
001, which gives a maximum Courant-Friedrichs-Levy number of
≈
0
.
15.
To determine the suitability of these parameters, we performed a grid convergence
study of the nonlinear solver using
Re
= 200
,M
ρ
= 0
.
5
,K
B
= 0
.
35. For these parameters
the flag enters limit cycle flapping of fixed amplitude and frequency. Using the grid
described above, the amplitude and frequency of these oscillations were
a
=
±
0
.
81
,f
=
0
.
180, respectively. Refining the grid spacing to
h
= 0
.
0075 on the finest domain and
increasing the domain such that the finest sub-domain size was [
−
0
.
2
,
2
.
8]
×
[
−
1
.
5
,
1
.
5]
8
A. Goza, T. Colonius, and J. Sader
and the total domain size was [
−
22
.
58
,
25
.
18]
×
[
−
23
.
88
,
23
.
88] changed these values to
a
=
±
0
.
80
,f
= 0
.
183, respectively.
3. Dynamics for
Re
= 200
We now consider the inverted-flag system for
Re
= 200. We demonstrate the existence
of a deformed equilibrium that is stable over a small range of stiffnesses and becomes
unstable as
K
B
is decreased. The transition to small-deflection deformed flapping as-
sociated with this decrease in
K
B
is shown through a global stability analysis to be a
supercritical Hopf bifurcation of the deformed equilibrium. We then consider the large-
amplitude flapping regime, and confirm the arguments of Sader
et al.
(2016
a
) that this
regime is a VIV for small values of
M
ρ
. VIV is shown to cease for larger mass ratios,
consistent with the scaling analysis of Sader
et al.
(2016
a
), and we demonstrate that
large-amplitude flapping persists despite the absence of VIV. The potential mechanisms
associated with this non-VIV large-amplitude flapping regime are discussed. We then
use a global stability analysis to confirm the argument of Shoele & Mittal (2016) that
small-amplitude flapping in the deflected-mode regime is driven by the bluff-body vortex-
shedding instability. Finally, we show that for a range of
K
B
, light flags with
M
ρ
6
O
(1)
exhibit chaotic flapping characterised by switching between large-amplitude flapping and
the deflected-mode state. No chaotic flapping is observed for heavy flags,
i.e.
,
M
ρ
> O
(1).
3.1.
Bifurcation diagrams and general observations
Figure 3 shows bifurcation diagrams at four different masses for
Re
= 200. Each plot
gives the transverse leading edge displacement (tip deflection,
δ
tip
, nondimensionalised by
the flag length
L
) as a function of the reciprocal stiffness (1
/K
B
). Solid lines represent
stable equilibria, and dashed lines correspond to unstable equilibria. Information for
unsteady regimes is conveyed through the markers. A set of markers at a given stiffness
corresponds to tip deflection values from a single nonlinear simulation at moments when
the tip velocity is zero (
i.e.
, when the flag changes direction at the tip). From a dynamical
systems perspective, the markers correspond to zero tip velocity Poincar ́e sections of a tip
velocity-tip displacement phase portrait. All nonlinear simulations were started with the
flag in its undeflected position and the flow impulsively started to its freestream value.
A small body force was introduced at an early time to trigger any instabilities in the
system. All simulations contain a minimum of 15 flapping cycles except for the chaotic
flapping regime, where a minimum of 55 cycles were used. To avoid representing transient
behaviour in the figures, we omit the first several flapping cycles in the diagrams. The
bifurcation diagrams were insensitive to starting conditions—the results were unchanged
by running a corresponding set of simulations with the flag initialised in its deformed
equilibrium state.
To illustrate the meaning of the markers in figure 3 further, consider 1
/K
B
≈
4 for
M
ρ
= 0
.
5. The system enters into large-amplitude limit cycle flapping with a fixed
amplitude of
≈ ±
0
.
8, and the bifurcation diagram reflects this with a marker at these
peak tip displacements, which are the only tip displacement values where the tip velocity
is zero. Note that there are actually several markers superposed onto one another at this
stiffness since multiple flapping periods were used to plot these diagrams, though only
one marker is visible because of the limit cycle behaviour exhibited. As another example,
the bifurcation diagram at 1
/K
B
≈
6 for
M
ρ
= 0
.
05 depicts chaotic flapping. Many
markers are visible at this stiffness because the flag changes direction at several different
values of
δ
tip
. The value of using zero tip-velocity Poincar ́e sections for the bifurcation
diagrams is seen through chaotic flapping: these Poincar ́e sections demonstrate the variety
Global modes and nonlinear analysis of inverted-flag flapping
9
Figure 3: Bifurcation diagrams of inverted-flag dynamics at
Re
= 200 that show
leading edge transverse displacement (tip deflection,
δ
tip
) versus inverse stiffness (1
/K
B
).
I: undeformed equilibrium, II: deformed equilibrium, III: small-deflection deformed
flapping, IV: large-amplitude flapping, V: chaotic flapping, VI: deflected mode. See the
main text for a description of the various lines and markers and details on how the
diagrams were constructed.
of transverse locations where the flag changes direction— a fact not captured through,
for example, plotting the peak-to-peak flapping amplitudes at a given stiffness.
The bifurcation diagrams in figure 3 depict the undeformed equilibrium (I), deformed
equilibrium (II), small-deflection deformed flapping (III), large-amplitude flapping (IV),
deflected mode (VI), and chaotic flapping (V) regimes. In small-deflection deformed flap-
ping, flapping is seen about the upward deflected equilibrium. There is a corresponding
deformed equilibrium with a negative flag deflection, and different initial conditions would
10
A. Goza, T. Colonius, and J. Sader
Figure 4: Vorticity contours for equilibrium states of the flow-inverted-flag system at
Re
= 20. From left to right:
K
B
= 0
.
5
,
0
.
35
,
0
.
22
,
0
.
11. The three rightmost equilibria are
unstable for all masses considered. Contours are in 18 increments from -5 to 5.
result in flapping about this equilibrium. We refrain from plotting this behaviour to avoid
confusion with large-amplitude flapping.
The undeformed equilibrium becomes unstable with decreasing stiffness due to a
divergence instability (the critical stiffness for instability is independent of the mass
ratio) (Kim
et al.
2013; Gurugubelli & Jaiman 2015; Sader
et al.
2016
a
). We see from
figure 3 that this instability causes a transition to a regime where the flag is in a steady
deflected position. As stiffness is decreased, this steady deflected state is characterised
by increasingly large tip deflections (see figure 4). This regime was first observed by Ryu
et al.
(2015); Gurugubelli & Jaiman (2015), and we note that it represents a deformed
equilibrium state (
i.e.
, in the notation of section 2 it satisfies the steady state equations
r
(
y
) = 0). Moreover, even for masses where flapping occurs, the deformed equilibrium
still exists as an unstable steady-state solution to the fully-coupled equations (2.5)–(2.8).
Note also that for a given stiffness the tip deflection of the deformed equilibrium is
constant for all masses, since the equilibrium is a steady state solution of (2.5)–(2.8)
and therefore does not depend on flag inertia. Figure 4 provides illustrations of deformed
equilibria for various stiffnesses (some of which are unstable).
Figure 5 gives the peak flapping frequency for the various regimes where flapping
occurs. For all masses considered, small-deflection flapping is associated with a low
frequency that is not indicative of VIV behaviour: using the maximal tip displacement as
the length scale, the largest Strouhal number over all masses is 0.02. We show in the next
section that this regime is caused by the transition to instability of the leading global
mode of the deformed equilibrium. Note that the frequency is dependent on flag mass in
this regime, illustrating the fully-coupled nature of the problem.
Large-amplitude flapping is qualitatively different for light flags (
M
ρ
= 0
.
05
,
0
.
5) and
heavy flags (
M
ρ
= 5
,
50).
Light flags
: the flapping frequency is roughly constant across an
order-of-magnitude change in
M
ρ
, which demonstrates the flow-driven nature of flapping
in this regime. Sader
et al.
(2016
a
) found that for a range of parameters large-amplitude
flapping exhibits several properties of a VIV, and we confirm below that for light flags
the fluid forces on the flag synchronise with the flag’s motion to form a VIV.
Heavy flags
:
the flapping frequency is decreased relative to light flags, and we show in a later section
that there is a corresponding de-synchronisation between flapping and vortex shedding.
Thus, for heavy flags large-amplitude flapping is not a VIV. This confirms the scaling
analysis of Sader
et al.
(2016
a
) that VIV behaviour should cease for sufficiently heavy
flags. We note from region IV of the bifurcation diagrams in figure 3 that for
M
ρ
= 5
,
50
large-amplitude flapping persist despite the absence of a VIV. We discuss the mechanism
for flapping in this non-VIV regime in section 3.3.
For all masses, the deflected-mode regime (occuring at low stiffness/ high flow rate)