Vortex shedding model of a flapping flag

doi:10.1017/S0022112008004321

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Vortex shedding model of a flapping flag

Journal Article•DOI•

Vortex shedding model of a flapping flag

Sébastien Michelin¹, Stefan G. Llewellyn Smith¹, Beverley J. Glover²•Institutions (2)

University of California, San Diego¹, University of Cambridge²

01 Dec 2008-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 617, pp 1-10

TL;DR: In this article, a two-dimensional model for the flapping of an elastic flag under axial flow is described and the vortical wake is accounted for by shedding of discrete point vortices with unsteady intensity, enforcing the regularity condition at the flag's trailing edge.

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Abstract: A two-dimensional model for the flapping of an elastic flag under axial flow is described. The vortical wake is accounted for by the shedding of discrete point vortices with unsteady intensity, enforcing the regularity condition at the flag's trailing edge. The stability of the flat state of rest as well as the characteristics of the flapping modes in the periodic regime are compared successfully to existing linear stability and experimental results. An analysis of the flapping regime shows the co-existence of direct kinematic waves, travelling along the flag in the same direction as the imposed flow, and reverse dynamic waves, travelling along the flag upstream from the trailing edge.

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Journal Article•DOI•

Flapping and Bending Bodies Interacting with Fluid Flows

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Michael Shelley¹, Jun Zhang¹•Institutions (1)

New York University¹

04 Jan 2011-Annual Review of Fluid Mechanics

TL;DR: In this paper, the authors review recent, highly detailed experiments that reveal new nonlinear phenomena in these systems, as well as advances in theoretical understanding, resulting in large part from the rapid development of new simulation methods that fully capture the mutual coupling of fluids and flexible solids.

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Abstract: The flapping or bending of a flexible planar structure in a surrounding fluid flow, which includes the flapping of flags and the self-streamlining of flexible bodies, constitutes a central problem in the field of fluid-body interactions. Here we review recent, highly detailed experiments that reveal new nonlinear phenomena in these systems, as well as advances in theoretical understanding, resulting in large part from the rapid development of new simulation methods that fully capture the mutual coupling of fluids and flexible solids.

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375 citations

Cites background from "Vortex shedding model of a flapping..."

...In a closely related approach, Michelin et al. (2008) have also developed a simpler, but fully nonlinear, model wherein the continuous vortex sheet shed by the flexible flag is replaced by the shedding of discrete point vortices with unsteady strengths....
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Journal Article•DOI•

Resonance and propulsion performance of a heaving flexible wing

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Sébastien Michelin, Stefan G. Llewellyn Smith

20 Jul 2009-Physics of Fluids

TL;DR: In this article, the influence of bending rigidity of a flexible heaving wing on its propulsive performance in a two-dimensional imposed parallel flow is investigated in the inviscid limit.

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Abstract: The influence of the bending rigidity of a flexible heaving wing on its propulsive performance in a two-dimensional imposed parallel flow is investigated in the inviscid limit. Potential flow theory is used to describe the flow over the flapping wing. The vortical wake of the wing is accounted for by the shedding of point vortices with unsteady intensity from the wing’s trailing edge. The trailing-edge flapping amplitude is shown to be maximal for a discrete set of values of the rigidity, at which a resonance occurs between the forcing frequency and a natural frequency of the system. A quantitative comparison of the position of these resonances with linear stability analysis results is presented. Such resonances induce maximum values of the mean developed thrust and power input. The flapping efficiency is also shown to be greatly enhanced by flexibility.

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221 citations

Cites background or methods from "Vortex shedding model of a flapping..."

...with wp the principal value of the relative tangential velocity on the wing [28]...
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...Taking advantage of the linear relation between [p]± and κ̇, added inertia terms can be isolated from the part of the pressure that can be explicitely computed at each time step, thereby avoiding the use of an iterative solver and greatly enhancing the computational efficiency [28]....
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...8 10−2 for μ = 2, using the same point vortex model [28])....
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...The fact that only odd harmonics appear in the tail motion was already previously observed in the case of a passive flag [28]....
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...(7) Equations (5)-(7) can be rewritten as a system for θ and T only [28]: Tss − θ(2) sT = −[p]θs − 2ηθsθsss − ηθ(2) ss − μθ̇(2) (8) μθ̈ = −[p]s − ηθssss + (T + ηθ(2) s)θss + 2Tsθs (9) θ(0, t) = θs(1, t) = θss(1, t) = T (1, t) = 0 (10)...
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Journal Article•DOI•

Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency

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Olivier Doaré¹, Sébastien Michelin²•Institutions (2)

Superior National School of Advanced Techniques¹, École Polytechnique²

01 Nov 2011-Journal of Fluids and Structures

TL;DR: In this article, the energy harvested from the flutter of a plate in an axial flow by making use of piezoelectric materials was investigated, and the equations for fully coupled linear dynamics of the fluid-solid and electrical systems were derived.

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197 citations

Cites background or result from "Vortex shedding model of a flapping..."

...Both local [44] and global [20] analyses of the nonlinear regime have been performed for compliant panels or plates placed in an axial flow....
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...In particular, the non-linear mode shape that determines the piezoelectric deformation rate and the energy transfers to the output circuit, has been observed to be similar to that of the most linearly unstable mode [20, 39]....
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Journal Article•DOI•

Energy harvesting efficiency of piezoelectric flags in axial flows

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Sébastien Michelin¹, Olivier Doaré²•Institutions (2)

École Polytechnique¹, Superior National School of Advanced Techniques²

10 Jan 2013-Journal of Fluid Mechanics

TL;DR: In this paper, the authors used piezoelectric patches attached to the surface of a flexible flag in axial flow to convert the solid deformation into an electric current powering purely resistive output circuits.

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Abstract: Self-sustained oscillations resulting from fluid–solid instabilities, such as the flutter of a flexible flag in axial flow, can be used to harvest energy if one is able to convert the solid energy into electricity. Here, this is achieved using piezoelectric patches attached to the surface of the flag, which convert the solid deformation into an electric current powering purely resistive output circuits. Nonlinear numerical simulations in the slender-body limit, based on an explicit description of the coupling between the fluid–solid and electric systems, are used to determine the harvesting efficiency of the system, namely the fraction of the flow kinetic energy flux effectively used to power the output circuit, and its evolution with the system’s parameters. The role of the tuning between the characteristic frequencies of the fluid–solid and electric systems is emphasized, as well as the critical impact of the piezoelectric coupling intensity. High fluid loading, classically associated with destabilization by damping, leads to greater energy harvesting, but with a weaker robustness to flow velocity fluctuations due to the sensitivity of the flapping mode selection. This suggests that a control of this mode selection by a careful design of the output circuit could provide some opportunities to improve the efficiency and robustness of the energy harvesting process.

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191 citations

Journal Article•DOI•

Energy harvesting efficiency of piezoelectric flags in axial flows

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Sébastien Michelin¹, Olivier Doaré²•Institutions (2)

École Polytechnique¹, Superior National School of Advanced Techniques²

08 Oct 2012-arXiv: Fluid Dynamics

TL;DR: In this article, the authors used nonlinear numerical simulations in the slender body limit, based on an explicit description of the coupling between the fluid-solid and electric systems, to determine the harvesting efficiency of the system, namely the fraction of the flow kinetic energy flux effectively used to power the output circuit.

...read moreread less

Abstract: Self-sustained oscillations resulting from fluid-solid instabilities, such as the flutter of a flexible flag in axial flow, can be used to harvest energy if one is able to convert the solid energy into electricity. Here, this is achieved using piezoelectric patches attached to the surface of the flag that convert the solid deformation into an electric current powering purely resistive output circuits. Nonlinear numerical simulations in the slender-body limit, based on an explicit description of the coupling between the fluid-solid and electric systems, are used to determine the harvesting efficiency of the system, namely the fraction of the flow kinetic energy flux effectively used to power the output circuit, and its evolution with the system's parameters. The role of the tuning between the characteristic frequencies of the fluid-solid and electric systems is emphasized, as well as the critical impact of the piezoelectric coupling intensity. High fluid loading, classically associated with destabilization by damping, leads to greater energy harvesting, but with a weaker robustness to flow velocity fluctuations due to the sensitivity of the flapping mode selection. This suggests that a control of this mode selection by a careful design of the output circuit could provide some opportunities of improvement for the efficiency and robustness of the energy harvesting process.

...read moreread less

181 citations

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Journal Article•DOI•

On The Instability Of Jets

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Lord Rayleigh

01 Nov 1878-Proceedings of The London Mathematical Society

2,819 citations

"Vortex shedding model of a flapping..." refers background in this paper

...The two-dimensional linear stability of infinite membranes under axial flow was first studied by Rayleigh (1878)....
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Journal Article•DOI•

Note on the swimming of slender fish

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M. J. Lighthill

01 Oct 1960-Journal of Fluid Mechanics

TL;DR: In this paper, the authors determine what transverse oscillatory movements a slender fish can make which will give it a high Froude propulsive efficiency, and the recommended procedure is for the fish to pass a wave down its body at a speed of around of the desired swimming speed, the amplitude increasing from zero over the front portion to a maximum at the tail, whose span should exceed a certain critical value.

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Abstract: The paper seeks to determine what transverse oscillatory movements a slender fish can make which will give it a high Froude propulsive efficiency, The recommended procedure is for the fish to pass a wave down its body at a speed of around of the desired swimming speed, the amplitude increasing from zero over the front portion to a maximum at the tail, whose span should exceed a certain critical value, and the waveform including both a positive and a negative phase so that angular recoil is minimized. The Appendix gives a review of slender-body theory for deformable bodies.

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1,090 citations

"Vortex shedding model of a flapping..." refers result in this paper

...A horizontal wave speed not greater than U∞ in this passive drag-producing configuration is consistent with the work of Lighthill (1960), where thrust production is associated with deformation waves travelling faster than U∞....
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Journal Article•DOI•

Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind

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Jun Zhang¹, Jun Zhang², Stephen Childress¹, Albert Libchaber², Michael Shelley¹ - Show less +1 more•Institutions (2)

New York University¹, Rockefeller University²

14 Dec 2000-Nature

TL;DR: The dynamics of swimming fish and flapping flags involves a complicated interaction of their deformable shapes with the surrounding fluid flow, and it is found that, for a single filament, there are two distinct, stable dynamical states.

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Abstract: The dynamics of swimming fish and flapping flags involves a complicated interaction of their deformable shapes with the surrounding fluid flow. Even in the passive case of a flag, the flag exerts forces on the fluid through its own inertia and elastic responses, and is likewise acted on by hydrodynamic pressure and drag. But such couplings are not well understood. Here we study these interactions experimentally, using an analogous system of flexible filaments in flowing soap films. We find that, for a single filament (or 'flag') held at its upstream end and otherwise unconstrained, there are two distinct, stable dynamical states. The first is a stretched-straight state: the filament is immobile and aligned in the flow direction. The existence of this state seems to refute the common belief that a flag is always unstable and will flap. The second is a flapping state: the filament executes a sinuous motion in a manner akin to the flapping of a flag in the wind. We study further the hydrodynamically coupled interaction between two such filaments, and demonstrate the existence of four different dynamical states.

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599 citations

"Vortex shedding model of a flapping..." refers background in this paper

...Hysteresis behaviour Experimental studies on flapping flags have pointed out the hysteresis behaviour of the flag when the velocity of the flow at infinity is varied (Zhang et al. 2000; Shelley et al. 2005; Eloy et al. 2008)....
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...Hysteresis behaviour Experimental studies on flapping flags have pointed out the hysteresis behaviour of the flag when the velocity of the flow at infinity is varied (Zhang et al. 2000; Shelley et al. 2005; Eloy et al. 2008)....
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...The fluttering flag instability has been the focus of a large number of experimental studies including soap-film (Zhang et al. 2000), water-tunnel (Shelley, Vandenberghe & Zhang 2005) and wind-tunnel experiments (Eloy et al. 2008)....
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...Future work will also include the study of two parallel flags as described in Zhang et al. (2000) to understand the coupling between the two flags and its influence on the stability and mode structure....
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...The fluttering flag instability has been the focus of a large number of experimental studies including soap-film (Zhang et al. 2000), water-tunnel (Shelley, Vandenberghe & Zhang 2005) and wind-tunnel experiments (Eloy et al....
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Journal Article•DOI•

Simulation of a Flapping Flexible Filament in a Flowing Soap Film by the Immersed Boundary Method

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Luoding Zhu¹, Charles S. Peskin¹•Institutions (1)

Courant Institute of Mathematical Sciences¹

01 Jul 2002-Journal of Computational Physics

TL;DR: In this paper, the simulation of a flapping flexible filament in a flowing soap film using the immersed boundary method is described. But the simulation is restricted to the case of a single filament.

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413 citations

"Vortex shedding model of a flapping..." refers methods in this paper

...Recently full numerical simulations of the coupled fluid and solid dynamics have been carried out using immersed-boundary methods (Zhu & Peskin 2002) and coupled fluid–solid solvers (Connell & Yue 2007)....
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...Recently full numerical simulations of the coupled fluid and solid dynamics have been carried out using immersed-boundary methods ( Zhu & Peskin 2002 ) and coupled fluid‐solid solvers (Connell & Yue 2007)....
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Journal Article•DOI•

Interaction of the Monsoon and Pacific Trade Wind System at Interannual Time Scales Part I: The Equatorial Zone

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Tim P. Barnett¹•Institutions (1)

University of California, San Diego¹

01 Apr 1983-Monthly Weather Review

TL;DR: The time history of the Monsoon System over the Indian Ocean has been developed from ship observations and merged with the Wyrtki-Meyers Pacific Trade Wind field by a rather new empirical orthogonal function (EOF) analysis capable of detecting propagating features in wind systems as mentioned in this paper.

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Abstract: The time history of the Monsoon System over the Indian Ocean has been developed from ship observations and merged with the Wyrtki-Meyers Pacific Trade Wind field. The interaction of these two massive wind systems has been studied by a rather new empirical orthogonal function (EOF) analysis capable of detecting propagating features in the wind systems. The current study (Part I) was confined to variations within ±10° of the equator. Results show the two wind systems are strongly coupled at interannual time scales. The coupling is effected through cyclostationary pulsations and longitudinal shifts of the huge surface convergence over Indonesia. The interaction may also he thought of as the spatial expansion/contraction of the wind systems. These changes can be viewed as the transition of the Monsoon/Trade Winds between two preferred climate states. One sub-element of this apparent bimodality in the wind fields is the El Nino phenomenon. The zonal component of the combined wind fields seems to insti...

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387 citations

"Vortex shedding model of a flapping..." refers methods in this paper

...The EOF decomposition is applied to the complex function F = f + if H where f H is the Hilbert transform of f (see Barnett 1983, for more details)....
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