Simple current source model neglecting piezoelectric feedback 

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Stability and dynamics of a single flag placed in a uniform flow

A great number of researchers have reported studies on stability and dynamics of a single flag placed in a uniform flow [Huang 1995, Eloy 2007, Shelley 2011]. In their experimental studies, Zhang et al. [Zhang 2000] observed the motion of a fil-ament in a 2D flow generated by a soap-film. They observed that depending on the length of the filament, it has two fundamental states: (i) the stretched-straight state when the length is small, and (ii) the flapping state when the length is large. They also observed the evolution of the flapping amplitude and frequency with the filament’s length, and identified a hysteresis phenomenon by increasing and reducing the filament’s length. Watanabe et al. investigated experimentally the paper flutter using different materials [Watanabe 2002b], and also developed theoretical models to account for various observations in the experiments [Watanabe 2002a]. In both studies, relations between the flutter speed and the mass ratio, i.e. the ratio between the inertia of the fluid and solid, are identified. Other experimental and theoreti-cal works also investigated the stability [Lemaitre 2005] and post-critical dynamics [Eloy 2012, Virot 2013, Gibbs 2014] of a flag placed in a wind tunnel and offered a vast catalogue of flutter properties of flags made of different materials, such as paper, plastic, fabrics, and metals. Recently, with the increasing capacity of scientific computing, a large amount of numerical work has been conducted to study the flapping of a flag. The ex-periments of Zhang et al. were reproduced by direct numerical simulation using immersed boundary method (IBM) [Zhu 2002] and arbitrary Lagrangian Eulerian (ALE) [Sawada 2006]. Both numerical methods gave results that are in a good qualitative agreement with the experimental results: (i) stretched-straight state is observed with short flags, (ii) bistability–switching between the stretched-straight state and flapping state–is observed for longer flags.
Despite the benefit that the direct numerical simulation provides a detailed de-scription of the dynamics of both the flow and the structure, it is time-consuming and prohibitive for high Reynolds numbers. As a result, many simplified models are developed to carry out faster simulations. A very popular model for describ-ing the dynamics of a flag is the inextensible Euler-Bernoulli beam model, while a variety of models for the fluid flow, based on the inviscid flow assumption and in-compressibility, are used by different researchers. Alben et al. used a flexible body vortex sheet model to compute the fluid forcing and the flow field around a flap-ping 2D flag [Alben 2008a]. They reported that in addition to the two previously mentioned states: i.e. the stretched-straight and periodic flapping states, a chaotic state, characterised by undefined amplitude and frequency, may appear when the incoming flow velocity is much larger than the critical velocity. Using a unsteady point vortex model, Michelin et al. also identified the existence of this chaotic state [Michelin 2008].
The experimental and numerical techniques for studying a 2D flag in a uniform 2D incoming flow are well developed and have been providing interesting insights of flag’s flapping dynamics. However, the major drawback of these techniques is that they consider flags of an infinite span, which is unrealistic. Many studies therefore also focus on 3D effects on the flapping flag. In their work, Eloy et al. highlighted that the flag’s span has a significant influence on the flag’s stability [Eloy 2007]: with a fixed flag’s length, the onset of flapping takes place at a lower velocity for a flag with larger span. This conclusion is supported by the study of Gibbs et al. [Gibbs 2012], in which experiments are performed and a stability analysis is carried out using the Euler-Bernoulli beam model to describe the flag, and a vortex lattice model [Tang 2007] to account for fluid loading. Another model for the fluid forcing, called Large-Amplitude Elongated-Body Theory (LAEBT), initially developed as to describe fish locomotion [Lighthill 1971, Candelier 2011], is recently adapted to the case of 3D flapping flags [Singh 2012b, Eloy 2012, Michelin 2013] by adding drag terms corresponding to the dissipation induced by the lateral flow separation due to the finite flag span. Meanwhile, DNS techniques are also developed for 3D simulations using both IBM [Tian 2012] and ALE [Bourlet 2015].
In many recent studies, a uniform, inviscid, incompressible flow and the Euler-Bernoulli beam model are used to investigate the flag’s flapping in a flow. Using these models, the system is controlled by three dimensionless parameters: the mass ratio M , the reduced velocity U , and the aspect ratio H . These parameters are defined as: U = U1Lr where sf and are respectively the fluid’s mass per unit surface and the flag’s mass per unit length, L is the length of the flag, U1 is the incoming flow velocity, B is the flag’s bending rigidity, and H is the flag’s span. Note that in the 3D case, sf = f H, with f representing the fluid’s density. Regardless of the various models used in different works, a widely approved conclusion is that for a flag with a given aspect ratio H (H = 1 in 2D cases), larger mass ratio M leads to lower critical velocity in terms of U , as shown in Fig. 1.4. When U < Uc , the flag is stable and stays in the stretched-straight state. Once U > Uc , the flag becomes unstable and reaches either a periodic flapping state (Fig. 1.5a, b), or a chaotic flapping state depending on the flow velocity (Fig. 1.5c, d).
Another aspect involved in the studies of a single flag in a uniform flow is the ef-fect of walls. Mainly three configurations are under active investigation: (i) the close presence of one single rigid wall parallel to the flag’s plane [Nuhait 2010, Dessi 2015], (ii) the close presence of two rigid walls parallel to the flag’s plane, thus forming a transverse confinement [Belanger 1995, Guo 2000, Alben 2015], and (iii) the close presence of two rigid walls orthogonal to the flag’s plane, therefore forming a spanwise confinement [Doaré 2011b, Doaré 2011c]. These studies showed that the pres-ence of one or two walls in the vicinity of the flag has a destabilising effect, i.e. the confinement reduces the critical velocity. Some studies also reported that the confinement leads to an increase of the flag’s added mass [Belanger 1995, Guo 2000]. Post-critical behaviour of a flag in a transverse confinement is also studied numer-ically by Alben [Alben 2015], who found that while decreasing the channel wall distance from infinity, the flapping amplitude starts by increasing, then decreases because it is limited by the wall. Note that in a wind tunnel test with a flag, both transverse and spanwise confinements may exist depending on the size of the wind tunnel’s test section.

Stability and dynamics of several flags placed in uniform flow

Studying the coupled motion of several flexible bodies placed in a flow is moti-vated by natural phenomena, particularly the fish schooling [Cushing 1968]. Weihs [Weihs 1973] pointed out that in a 2D plane, the optimal positioning of each fish in a school should have a diamond pattern (Fig. 1.6) so that the fish that follow others would profit from the thrust induced by the oscillatory motion of their predecessors.
The constantly improving techniques for studying a single flag’s flapping are providing new methods to fulfil researchers’ motivation in studying the flapping of multiple flags. Zhang et al. conducted experiments using two filaments placed side by side in a 2D flow based on a soap film [Zhang 2000]. Their results show that under the same incoming flow, two filaments flap in an in-phase pattern (two flags have the same vertical displacement) when the distance separating them is small. The flapping becomes out of phase (two flags have the opposite vertical displace-ment) when the two filaments are moved away from each other. The observation of the two flapping patterns, i.e. the out-of-phase one and the in-phase one, is also reproduced by numerical simulations of parallel 2D flags [Zhu 2003, Farnell 2004]. Using the vortex sheet model and the Euler-Bernoulli beam, Alben [Alben 2009b] reported that the phase difference between two side-by-side 2D flags evolves almost monotonically with the distance separating them.
Jia et al. [Jia 2007] performed more thorough experimental investigations of two identical side-by-side filaments placed in a flowing soap film, and studied theoreti-cally their linear stability. They suggest that the coupled dynamics of two filaments is subject to three dimensionless parameters: the mass ratio M , the reduced veloc-ity U , and the dimensionless form of the separation distance d, defined by: d = D ; (1.2) where D is the dimensional form of the separation distance. According to the vari-ation of these parameters, four flapping modes may be identified: (i) the stretched-straight mode, (ii) in-phase mode, (iii) out-of-phase mode, and (iv) an indefinite mode, i.e. a phase difference switching between 0 and . Using a double-wake model, Michelin & Llewellyn Smith [Michelin 2009] studied the linear stability of two side-by-side flags and observed that a decreasing d induces to a destabilising ef-fect, thereby lowering the critical velocity Uc . Wang et al. [Wang 2010] performed wind tunnel tests with two identical flags placed side by side and confirmed this destabilisation. In addition, they found that when d is too small (d < 0:2), Uc ac-tually becomes much higher than the critical velocity of one single flag. They argue that the very small d actually makes the two flags to behave as one single flag of a larger thickness, thus a higher flow velocity is required to destabilise the system.
Another basic configuration involving two flags placed in tandem is also studied.
Still using filaments in soap-film flow, Jia & Yin [Jia 2008] performed experiments with two filaments placed in tandem, i.e. one filament is placed directly down-stream to the other. They investigated flapping patterns and the energy distribu-tion of two filaments by varying the separation distance. Their results show that the downstream filament experiences a drafting induced by the wake of the upstream filament. As a result of the drafting, the drag force on the downstream filament is reduced. Ristroph & Zhang [Ristroph 2008] also conducted experimental work using a soap film and two filaments placed in tandem but found a result opposite to Jia & Yin: the upstream filament actually experiences an inverted drafting. The drag applied on the upstream flag is lower than the drag on the downstream flag. The reason that they found opposite results lies probably in the different leading edge conditions used in these two studies: in [Jia 2008], the upstream filament is fixed at its leading edge, while the leading edge of the downstream filament is tethered by a silk fibre fixed at the other end; in [Ristroph 2008], both flags have a fixed leading edge. The inverted drafting is confirmed by Alben [Alben 2009b] using a vortex sheet model, while Kim et al. [Kim 2010], using an improved version of IBM, reported that both drafting and inverted drafting can be observed depending on the phase difference of both flags’ vortex shedding.
During the last decade, an increasing number of researchers are interested in the coupled dynamics of three or more flexible bodies in uniform flow. Schouveiler et al. [Schouveiler 2009] performed wind tunnel tests with three and four side-by-side flags. Their work reported three possible flapping modes of three flags: (i) in-phase mode, i.e. all three flags have the same vertical motion, (ii) out-of-phase mode, i.e. two consecutive flags have opposite vertical motions, and (iii) symmetrical mode, i.e. the flag in the middle is stretched-straight, while the other two have opposite vertical motions. Michelin & Llewellyn Smith [Michelin 2009] extended the double-wake method to three and more side-by-side flags and investigated their linear stability: for the case of three flags, they also found the three modes reported in [Schouveiler 2009], and for the case of an infinite number of flags, the out-of-phase mode is found to be the dominant one for small M and large d, while for other parameters, the authors reported the existence of modes with the phase difference of any value between 0 and .

Concluding remarks: why we choose piezoelectric flags

The choice of the flag’s flutter instability as an energy-harvesting mechanism is motivated by its periodic, large-amplitude post-critical motion, which is the main feature of the the flag’s flapping dynamics. Such motion involves a permanent energy exchange between the flags and the surrounding fluid. Many researchers are getting interested in this energy exchange and are seeking ways to harvest energy from it. In general, energy harvesting based on flapping flags may follow two routes: producing energy either from the displacement [Tang 2008, Virot 2015] or from the deformation of the flag [Allen 2001, Singh 2012a]. The latter route has recently been the focus of several studies based on active materials [Doaré 2011a, Dunnmon 2011, Giacomello 2011, Akcabay 2012, Michelin 2013]. In our work, we will be interested in piezoelectric materials. The piezoelectric material is chosen in this work for its property of converting a part of mechanical energy generated from mechanical deformation to the electrical energy. In the next section, a brief introduction to piezoelectricity will be presented.

A brief introduction to piezoelectricity

Piezoelectric materials, as the name indicates, give an “electric” response under “pres-sure”. More precisely, such materials produce electric charge displacement when they are deformed. The discovery of piezoelectricity is attributed to Jacques & Pierre Curie [Curie 1880a, Curie 1880b]. However, their work in 1880 only revealed one piezoelectric effect, the effect that generates electric charge from the material’s defor-mation, since called direct piezoelectric effect. The other effect, called inverse piezoelectric effect remained in shadow at that time until one year later another French physicist, Gabriel Lippmann, who announced that according to the princi-ple of electric charge conservation, a piezoelectric crystal should experience a slight deformation under the influence of an external electric field [Lippmann 1881], a con-clusion that, although based solely on mathematical arguments, was experimentally confirmed in the same year by Jacques & Pierre Curie [Curie 1881].
In 1880, the Curie brothers published their work with the following statement [Curie 1880b]: Quelle que soit la cause déterminante, toutes les fois qu’un cristal hémiè-dre à faces inclinées, non conducteur, se contracte, il y a formation de pôles électriques dans un certain sens; toutes les fois que ce cristal se dilate, le dégagement d’électricité a lieu en sens contraire.
In this paragraph written in French, the Curie brothers, studying only the quartz, a naturally piezoelectric crystal, reached the conclusion that the reason of electric charge generation under deformation is the formation of electric dipoles within the material itself. This conclusion applies to almost all known piezoelectric materials, though the origin of electrical dipoles varies according to the specific category where a material lies [WEB2 , Ramadan 2014].
Knowing that electrical dipoles are the origin of piezoelectricity, we are able to describe in a qualitative way how the direct and inverse piezoelectric effects are produced. On one hand, without any deformation, a piezoelectric material is electri-cally neutral, implying that its centres of both positive and negative charges coincide. These two centres would be separated if the material is stretched, compressed, or sheared, creating an electrical field within the material, therefore a voltage difference is generated. An electric charge displacement would occur if the material’s positive side and negative side are connected with a conductive wire, hence the direct piezo-electric effect. On the other hand, an externally applied electric field would disrupt the electrical neutrality of a piezoelectric material. In order to restore the neutrality, the initially coinciding positive and negative centres would repulse each other as to create an electric field that compensates the external one. As a result, a mechan-ical deformation occurs, and an additional stress is induced due to the material’s elasticity, hence the inverse piezoelectric effect.
Take the elementary crystal structure of PZT (Lead zirconate titanate) for ex-ample. Figure 1.7a shows that when the structure is neutral, its centres of positive and negative charges are at the same point. When a mechanical strain on the x direction is applied on the structure (Fig. 1.7b), these two centres are moved away from each other, in the y direction, thus creating an electric field directing from the positive centre to the negative centre. If an external electric field following the increasing y direction is applied on the structure, in order to restore the electric neu-trality within the structure, the centres of positive and negative charges will move towards opposite directions, inducing consequently a deformation of structure.
Existing piezoelectric materials can fall into four categories: quartz, ceramics, polymers, and composites [Vijaya 2012].
wise confinement [Doaré 2011b, Doaré 2011c]. These studies showed that the pres-ence of one or two walls in the vicinity of the flag has a destabilising effect, i.e. the confinement reduces the critical velocity. Some studies also reported that the confinement leads to an increase of the flag’s added mass [Belanger 1995, Guo 2000]. Post-critical behaviour of a flag in a transverse confinement is also studied numer-ically by Alben [Alben 2015], who found that while decreasing the channel wall distance from infinity, the flapping amplitude starts by increasing, then decreases because it is limited by the wall. Note that in a wind tunnel test with a flag, both transverse and spanwise confinements may exist depending on the size of the wind tunnel’s test section.

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Stability and dynamics of several flags placed in uniform flow

Studying the coupled motion of several flexible bodies placed in a flow is moti-vated by natural phenomena, particularly the fish schooling [Cushing 1968]. Weihs [Weihs 1973] pointed out that in a 2D plane, the optimal positioning of each fish in a school should have a diamond pattern (Fig. 1.6) so that the fish that follow others would profit from the thrust induced by the oscillatory motion of their predecessors.
The constantly improving techniques for studying a single flag’s flapping are providing new methods to fulfil researchers’ motivation in studying the flapping of multiple flags. Zhang et al. conducted experiments using two filaments placed side by side in a 2D flow based on a soap film [Zhang 2000]. Their results show that under the same incoming flow, two filaments flap in an in-phase pattern (two flags have the same vertical displacement) when the distance separating them is small. The flapping becomes out of phase (two flags have the opposite vertical displace-ment) when the two filaments are moved away from each other. The observation of the two flapping patterns, i.e. the out-of-phase one and the in-phase one, is also reproduced by numerical simulations of parallel 2D flags [Zhu 2003, Farnell 2004]. Using the vortex sheet model and the Euler-Bernoulli beam, Alben [Alben 2009b] reported that the phase difference between two side-by-side 2D flags evolves almost monotonically with the distance separating them.
Jia et al. [Jia 2007] performed more thorough experimental investigations of two identical side-by-side filaments placed in a flowing soap film, and studied theoreti-cally their linear stability. They suggest that the coupled dynamics of two filaments is subject to three dimensionless parameters: the mass ratio M , the reduced veloc-ity U , and the dimensionless form of the separation distance d, defined by: d = D ; (1.2)
where D is the dimensional form of the separation distance. According to the vari-ation of these parameters, four flapping modes may be identified: (i) the stretched-straight mode, (ii) in-phase mode, (iii) out-of-phase mode, and (iv) an indefinite mode, i.e. a phase difference switching between 0 and . Using a double-wake model, Michelin & Llewellyn Smith [Michelin 2009] studied the linear stability of two side-by-side flags and observed that a decreasing d induces to a destabilising ef-fect, thereby lowering the critical velocity Uc . Wang et al. [Wang 2010] performed wind tunnel tests with two identical flags placed side by side and confirmed this destabilisation. In addition, they found that when d is too small (d < 0:2), Uc ac-tually becomes much higher than the critical velocity of one single flag. They argue that the very small d actually makes the two flags to behave as one single flag of a larger thickness, thus a higher flow velocity is required to destabilise the system.
Still using filaments in soap-film flow, Jia & Yin [Jia 2008] performed experiments with two filaments placed in tandem, i.e. one filament is placed directly down-stream to the other. They investigated flapping patterns and the energy distribu-tion of two filaments by varying the separation distance. Their results show that the downstream filament experiences a drafting induced by the wake of the upstream filament. As a result of the drafting, the drag force on the downstream filament is reduced. Ristroph & Zhang [Ristroph 2008] also conducted experimental work using a soap film and two filaments placed in tandem but found a result opposite to Jia & Yin: the upstream filament actually experiences an inverted drafting. The drag applied on the upstream flag is lower than the drag on the downstream flag. The reason that they found opposite results lies probably in the different leading edge conditions used in these two studies: in [Jia 2008], the upstream filament is fixed at its leading edge, while the leading edge of the downstream filament is tethered by a silk fibre fixed at the other end; in [Ristroph 2008], both flags have a fixed leading edge. The inverted drafting is confirmed by Alben [Alben 2009b] using a vortex sheet model, while Kim et al. [Kim 2010], using an improved version of IBM, reported that both drafting and inverted drafting can be observed depending on the phase difference of both flags’ vortex shedding.
During the last decade, an increasing number of researchers are interested in the coupled dynamics of three or more flexible bodies in uniform flow. Schouveiler et al. [Schouveiler 2009] performed wind tunnel tests with three and four side-by-side flags. Their work reported three possible flapping modes of three flags: (i) in-phase mode, i.e. all three flags have the same vertical motion, (ii) out-of-phase mode, i.e. two consecutive flags have opposite vertical motions, and (iii) symmetrical mode, i.e. the flag in the middle is stretched-straight, while the other two have opposite vertical motions. Michelin & Llewellyn Smith [Michelin 2009] extended the double-wake method to three and more side-by-side flags and investigated their linear stability: for the case of three flags, they also found the three modes reported in [Schouveiler 2009], and for the case of an infinite number of flags, the out-of-phase mode is found to be the dominant one for small M and large d, while for other parameters, the authors reported the existence of modes with the phase difference of any value between 0 and .

Concluding remarks: why we choose piezoelectric flags

The choice of the flag’s flutter instability as an energy-harvesting mechanism is motivated by its periodic, large-amplitude post-critical motion, which is the main feature of the the flag’s flapping dynamics. Such motion involves a permanent energy exchange between the flags and the surrounding fluid. Many researchers are getting interested in this energy exchange and are seeking ways to harvest energy from it. In general, energy harvesting based on flapping flags may follow two routes: producing energy either from the displacement [Tang 2008, Virot 2015] or from the deformation of the flag [Allen 2001, Singh 2012a]. The latter route has recently been the focus of several studies based on active materials [Doaré 2011a, Dunnmon 2011, Giacomello 2011, Akcabay 2012, Michelin 2013]. In our work, we will be interested in piezoelectric materials. The piezoelectric material is chosen in this work for its property of converting a part of mechanical energy generated from mechanical deformation to the electrical energy. In the next section, a brief introduction to piezoelectricity will be presented.

A brief introduction to piezoelectricity

Piezoelectric materials, as the name indicates, give an “electric” response under “pres-sure”. More precisely, such materials produce electric charge displacement when they are deformed. The discovery of piezoelectricity is attributed to Jacques & Pierre Curie [Curie 1880a, Curie 1880b]. However, their work in 1880 only revealed one piezoelectric effect, the effect that generates electric charge from the material’s defor-mation, since called direct piezoelectric effect. The other effect, called inverse piezoelectric effect remained in shadow at that time until one year later another French physicist, Gabriel Lippmann, who announced that according to the princi-ple of electric charge conservation, a piezoelectric crystal should experience a slight deformation under the influence of an external electric field [Lippmann 1881], a con-clusion that, although based solely on mathematical arguments, was experimentally confirmed in the same year by Jacques & Pierre Curie [Curie 1881].
In 1880, the Curie brothers published their work with the following statement [Curie 1880b]: Quelle que soit la cause déterminante, toutes les fois qu’un cristal hémiè-dre à faces inclinées, non conducteur, se contracte, il y a formation de pôles électriques dans un certain sens; toutes les fois que ce cristal se dilate, le dégagement d’électricité a lieu en sens contraire.
In this paragraph written in French, the Curie brothers, studying only the quartz, a naturally piezoelectric crystal, reached the conclusion that the reason of electric charge generation under deformation is the formation of electric dipoles within the material itself. This conclusion applies to almost all known piezoelectric materials, though the origin of electrical dipoles varies according to the specific category where a material lies [WEB2 , Ramadan 2014].
Knowing that electrical dipoles are the origin of piezoelectricity, we are able to describe in a qualitative way how the direct and inverse piezoelectric effects are produced. On one hand, without any deformation, a piezoelectric material is electri-cally neutral, implying that its centres of both positive and negative charges coincide. These two centres would be separated if the material is stretched, compressed, or sheared, creating an electrical field within the material, therefore a voltage difference is generated. An electric charge displacement would occur if the material’s positive side and negative side are connected with a conductive wire, hence the direct piezo-electric effect. On the other hand, an externally applied electric field would disrupt the electrical neutrality of a piezoelectric material. In order to restore the neutrality, the initially coinciding positive and negative centres would repulse each other as to create an electric field that compensates the external one. As a result, a mechan-ical deformation occurs, and an additional stress is induced due to the material’s elasticity, hence the inverse piezoelectric effect.
Take the elementary crystal structure of PZT (Lead zirconate titanate) for ex-ample. Figure 1.7a shows that when the structure is neutral, its centres of positive and negative charges are at the same point. When a mechanical strain on the x direction is applied on the structure (Fig. 1.7b), these two centres are moved away from each other, in the y direction, thus creating an electric field directing from the positive centre to the negative centre. If an external electric field following the increasing y direction is applied on the structure, in order to restore the electric neu-trality within the structure, the centres of positive and negative charges will move towards opposite directions, inducing consequently a deformation of structure.
Existing piezoelectric materials can fall into four categories: quartz, ceramics, polymers, and composites [Vijaya 2012].

Table of contents :

1 Introduction 
1.1 Overview of flow energy harvesting
1.2 Flutter instability
1.2.1 Stability and dynamics of a single flag placed in a uniform flow 3
1.2.2 Stability and dynamics of several flags placed in uniform flow 6
1.2.3 Concluding remarks: why we choose piezoelectric flags
1.3 A brief introduction to piezoelectricity
1.4 Energy harvesting using piezoelectric materials
1.5 Introduction of numerical models used in the present work
1.5.2 Piezoelectric effects
1.5.3 Dimensionless equations
1.6 Energy harvesting
1.7 Energy harvesting using piezoelectric flag connected to resistive circuits
1.8 Outline of manuscript
2 Single Piezoelectric Coverage 
2.1 Experimental set-up
2.2 Comparison between PVDF and MFC
2.3 Modelling of a flag covered by one piezoelectric pair
2.3.1 Simple current source model neglecting piezoelectric feedback
2.3.2 Nonlinear numerical model
2.4 Characterisation of the coupling coefficient
2.4.1 Measurement of B
2.4.2 Measurement of
2.5 Experimental and numerical results
2.5.1 PVDF flag in Tunnel A
2.5.2 PVDF flag in Tunnel B
2.5.3 MFC flag and feedback of piezoelectric effect
2.6 Summary and conclusion
3 Fluid-solid-electric lock-in 
3.1 Modelling of a flag continuously covered by pairs of piezoelectric patches
3.2 Linear stability
3.3 Nonlinear dynamics and energy harvesting
3.4 Impact of the coupling factor
3.5 Perspective: lock-in with one single piezoelectric pair
3.6 Conclusion
4 Non-local electric network 
4.1 Equations of non-local electric network
4.1.1 Flag covered by a finite number of piezoelectric pairs
4.1.2 Periodic networks and continuous limit
4.1.3 Boundary conditions and energy balance
4.1.4 Dimensionless Equations
4.2 Purely resistive circuits
4.3 Purely inductive circuits
4.3.1 Frequency lock-in
4.3.2 Energy harvesting at ext!0 1
4.4 Electrical energy flux
4.5 Conclusion and perspectives
5 Coupled flutter 
5.1 Two piezoelectric flags connected in one circuit
5.1.1 Electrical circuits
5.1.2 Harvesting efficiency
5.2 Fluid forcing: vortex sheet model
5.3 Side-by-side flags
5.3.1 Influence of in-phase and out-of-phase flapping
5.3.2 Influence of separation distance d
5.3.3 Resistive-inductive circuits
5.4 Flags in tandem
5.4.1 Resistive circuit
5.4.2 Resistive-inductive circuit
5.5 Conclusion and perspectives
6 Conclusion and perspectives 
6.1 Conclusion
6.1.1 Frequency lock-in
6.1.2 Two flags’ synchronisation through the fluid-solid-electric resonance
6.2 Perspectives
6.2.1 External forcing-induced vibration of piezoelectric flag
6.2.2 Flags placed in other types of flows
6.2.3 Flags positioned in alternative configurations
Appendices

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