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Contributed by Peter G. Schultz, December 17, 2004

We give an explanation for the onset of fluid-flow-induced flutter in a flag. Our theory accounts for the various physical mechanisms at work: the finite length and the small but finite bending stiffness of the flag, the unsteadiness of the flow, the added mass effect, and vortex shedding from the trailing edge. Our analysis allows us to predict a critical speed for the onset of flapping as well as the frequency of flapping. We find that in a particular limit corresponding to a low-density fluid flowing over a soft high-density flag, the flapping instability is akin to a resonance between the mode of oscillation of a rigid pivoted airfoil in a flow and a hinged-free elastic plate vibrating in its lowest mode.

The flutter of a flag in a gentle breeze and the flapping of a sail in a rough wind are commonplace and
familiar observations of a rich class of problems involving the interaction of fluids and structures, of
wide interest and importance in science and engineering (

Physically, the meaning of this result is as follows: For a heavy flag in a rapid flow, the added mass
effect due to fluid motion is negligible so that the primary effect of the fluid is an inertial pressure
forcing on the plate. For a plate of length _{f}U^{2}θ, where
ρ_{f}_{s}_{a}_{
s}h^{2}_{a}L_{a}_{f}U^{2}
/ρ_{s}hL^{1/2}. On the other hand, for a flexible plate of
thickness ^{3}) the elastic restoring force per unit length due to a deflection by
the same angle θ scales as ^{3}θ/^{3}
so that the frequency of the lowest mode of free bending vibrations of a flexible plate ω_{b}^{2}/ρ_{s}L^{4})^{1/2}. Equating the two yields a critical velocity for the onset of flutter of a plate
of given length _{c}^{3}/ρ_{f}L^{3})^{1/2}. As we will see in the following sections, this simple
result arises naturally from the analysis of the governing equations of motion of the flag and the fluid. In
particular, our analysis is capable of accounting for the unsteady nature of the problem in terms of the
added mass of the fluid and the vortex shedding from the trailing edge in terms of the seminal ideas of
Theodorsen (

We consider the dynamics of an inextensible two-dimensional elastic plate_{s}_{f}_{b}_{s}hl^{3}^{2}) is its flexural rigidity (here σ is the Poisson ratio of the material),
Δ

Schematic representation of the system. An elastic plate of length

In deriving Eq.

To close the system of Eqs.

We will assume that the flow is incompressible, inviscid, and irrotational. Then the tension in the
flag _{nc}, and a
circulatory potential, φ_{γ}, with φ = φ_{nc}
_{γ}. Then φ satisfies the Laplace equation, ∇^{
2}φ = 0, characterizing the two-dimensional fluid velocity field, (_{x}, φ_{y}
_{Y=0} = _{t}_{x}

For small deflections of the plate, the transverse velocity of the fluid, _{t}
_{x}_{
xt}) (and higher) that correspond physically to the rate of change of the local
angle of the plate, which can only be systematically accounted for in a non-local way._{tt}_{xt}

Kelvin's theorem demands that vorticity is conserved in an inviscid flow of given topology. Thus, the
circulatory flow associated with vortex shedding from the trailing edge requires a vorticity
distribution in the wake of the airfoil and a (bound) vorticity distribution in the airfoil to conserve
the total vorticity. If a point vortex shed from the trailing edge of the plate with strength
–Γ has a position (_{0}), _{0} ≥ 1, we must add a point vortex of strength Γ in the interior of
the sheet at (_{0})). This leads to a circulatory
velocity potential along the plate (_{0} = ((_{0} + 1/_{0})/2) characterizes the nondimensional _{0})/2). Therefore, for a distribution of vortices of strength γ defined by Γ =
γ(_{0}, the circulatory velocity potential is _{t}
_{γ} = (2/_{
x}_{0} φ_{γ}._{0} – (2/_{nc}_{γ}, i.e., _{t}_{x}

Substituting Eq. _{B}_{f}L_{f}_{s}_{0} = (_{b}_{
τ} + _{0}η_{s}_{s}_{τ} arises
because the plate exchanges momentum with the fluid, so that the time reversibility τ →
–τ symmetry is also broken. These two leading terms in the pressure, which could have
been written down on grounds of symmetry, correspond to a lift force proportional to η_{s}_{τ}. By
considering the detailed physical mechanisms, we find that the actual form of these terms is more
complicated due to the inhomogeneous dimensionless functions _{0}.

Since the free vortex sheet is advected with the flow, the vorticity distribution may be written as
γ = γ((2_{1}) – _{0}) and _{1}
being the time at which shedding occurs, which in dimensionless terms reads γ =
γ(2_{0}(τ – τ_{1}) – _{0}). Accounting for the oscillatory nature of the flapping instability with an
unknown frequency ω suggests that an equivalent description of the vorticity distribution is
given by γ = ^{i}^{(ω(τ–τ1)–qxo}), where _{0} is a nondimensional wave number of the vortex
sheet. Using the above traveling wave form of the vorticity distribution in Eq. _{i}

To understand the mechanism of instability of the trivial solution of _{0} ≫ 1. We can thus simplify Eq. _{0} → ∞ but with the scaled aerodynamic pressure _{t}_{x}
^{3}
η/^{4}) with the aerodynamic forces ρ_{f}
U^{2}(η/_{0} ∼
1/ρ^{1/2}. Then the typical flapping frequency ω is given by balancing plate
inertia ρ_{s}h^{2}η with the aerodynamic forces
ρ_{f}U^{2}(η/^{
στ} and compute the associated spectrum, σ(_{0})
using _{0} ≥ _{0}, the four eigenvalues with
smallest absolute value collide and split, leading to an instability via a 1:1 resonance (

Spectrum _{0} = 0.9_{0} (disks) and for _{0} = 1.1_{0} (square). We see that instability
occurs via a collision and splitting of two pairs of eigenvalues along the imaginary
axis (indicated by the arrows) and is a signature of a 1:1 resonance mechanism in a
time-reversible system.

As ρ ∼ _{0}_{τ}, becomes important, so
that the spectrum is shifted to the left, i.e., ^{στ} into Eq. _{0}) depends on
ω. We again solve the resulting system numerically with the _{0}) = _{0} > _{0}(ρ), _{0}, the plate is always unstable, i.e., large
enough fluid velocities will always destabilize the elastic plate. As ρ ≫ 1, the added
mass effect becomes relatively more important and it is easier for the higher modes of the plate to be
excited. We note that the stability boundary when _{0} than
that obtained by using the quasi-steady approximation

Stability diagram obtained by solving the eigenvalue problem _{0}. The thin dashed line represents the transition curve using the
quasi-steady approximation where _{0} below this line, the flag is stable, and for values above it, it is
unstable. The thin solid line represents the transition curve when vortex
shedding is taken into account, i.e., ^{
4}. The dots correspond to experimental data characterizing the transition
to flutter in three-dimensional flows past flexible sheets of paper (_{0}) as a function of ρ. When ρ ≪ 1,

Snapshots corresponding to the mode of instability ξ_{(}
_{s}
_{)} with ρ = 0.2, _{0} ≈ 66 (_{0} ≈ 6.6 (

In

Our linearized theory cannot capture the bistability in the transition to flutter without accounting
for the various possible nonlinearities in the system arising from geometry. But even without accounting
for these nonlinearities, there is a systematic discrepancy between our theory and the data, which
consistently show a higher value of _{0} for the onset of the instability. While
there are a number of possible reasons for this, we believe that there are three likely candidates: The
role of nonlocal interactions, three-dimensional effects, and the tension in the plate induced by the
Blasius boundary layer, all of which would tend to stabilize the sheet and thus push the onset to higher
values of _{0}. Postponing the question of nonlocal interactions to the future,
in ^{4}, making the comparison with the experimental data
quantitatively better. A final remark concerns the role of the free vortex sheet behind the flag. We have
assumed that this sheet is localized to the vicinity of

The commonplace occurrence of flutter in a flag belies the complexity hidden in this phenomenon. Extracting a qualitative and quantitative understanding involves the consideration of a number of effects. Our hierarchy of models starting with a relatively simple physical picture of the basic resonance-like behavior to the more sophisticated analyses in the quasi-steady and the unsteady cases have allowed us to dissect the physical mechanisms involved. In particular, we account for the finite length and finite bending stiffness of the sheet, as well as the fluid effects due to added-mass, vortex shedding, three- dimensional flow, and viscous boundary layer drag. We also provide a relatively simple criteria for the onset of the instability in terms of the scaling laws (19, 20). There are clear avenues for further questions, the most prominent of which include a detailed comparison with a two-dimensional numerical simulation and further experiments; these will be reported elsewhere.

This work was begun and brought to fruition when we were at Cambridge University; we dedicate this paper to the memory of Professor David G. Crighton, who first interested one of us (L.M.) in this problem nearly a decade ago. We thank M. Shelley, N. Vandenburghe, and J. Zhang for sharing their preprint with us. This work was supported by the European Community through Marie-Curie Fellowship HPMF-2002-01915 (to M.A.) and the U.S. Office of Naval Research through a Young Investigator Award (to L.M.).

Our analysis also carries over to the case of an elastic filament in a two-dimensional parallel flow.

In the appendices, we treat the case where

The general solution of the Laplace equation in two dimensions with the given boundary conditions may
be written as φ = ∫ _{t}
_{x}_{
t}_{x}

When the plate moves, fluid must also be displaced and the sheet behaves as if it had more inertia
(

This implies a neglect of any acceleration phase of the vorticity, a reasonable assumption at high

This is tantamount to the statement that that the inclusion of viscosity, no matter how small, will regularize the flow in the vicinity of the trailing edge.

This is because the term breaking time reversal symmetry ρ_{0}η_{τ} becomes negligibly small.

In this appendix, we introduce an approach that accounts for three-dimensional effects. By
introducing the noncirculatory velocity potential (Eq. _{nc}
^{+}, ^{+}). More generally, we define the function _{nc}
^{+}) is the noncirculatory potential corresponding to the
infinitely wide flag. _{nc}
^{+} as a function of

Three-dimensional effects. (

In high Reynolds number flows past the flag, the viscous boundary layer exerts a shear stress on the
flag that puts in under a variable tension _{x}^{4} shifts the marginal stability curve upwards in
the direction of the experimentally obtained one, as shown in

Quantitation of ceramide (Cer), hexosylceramide (HexosylCer),
lactosylceramide (LactosylCer), SM4s, and SM3 in lipid extracts from two brains
from WT mice (WT 1 and 2) and two brains from

Sphingolipid | WT 1 | WT 2 | ||
---|---|---|---|---|

Cer | 1,200 | 610 | 730 | 300 |

HexosylCer | 4,400 | 4,700 | 4,700 | 7,600 |

LactosylCer | <200 | <200 | 3,700 | 2,700 |

SM4s | 2,200 | 2,000 | 2,800 | 2,700 |

SM3
(LacCer II^{3}-sulfate) |
<40 | <40 | 450 | 480 |

Values are in picomoles of lipid per milligram of wet weight.