Linear stability analysis of coupled parallel flexible plates in an axial flow

Transcription

1 Linear stability analysis of couple parallel flexible plates in an axial flow Sébastien Michelin,a,b, Stefan G. Llewellyn Smith a a Department of Mechanical an Aerospace Engineering, Jacobs School of Engineering, UCSD, 9 Gilman Drive, La Jolla CA 993-, USA b Ecole Nationale Supérieure es Mines e Paris, Boulevar Saint Michel, 77 Paris ceex,france Abstract We stuy here the linear stability of N ientical flexible plates with clampe-free bounary conitions force by a uniform parallel flow. Flow viscosity an elastic amping are neglecte, an the flow aroun the plates is assume potential. The sheing of vorticity from the plates trailing eges is accounte for by introucing a force-free wake behin each plate. A Galerkin metho is use to compute the eigenmoes of the system. We are intereste in the effects of the number of plates an their relative istance on the stability property of the state of rest, as well as in the nature an structure of the couple states. Detaile results are presente for the cases N =, N = 3 an N. Keywors: Flow-inuce vibration, Flutter instability, Plate-assemblies PACS:.7.De,..-f, 7..Cq. Introuction The flapping motion of flexible plates or flags place in a parallel impose flow has been the subject of a large number of both experimental (Zhang et al., ; Shelley et al., ) an numerical stuies (Connell an Yue, 7; Zhu an Peskin, ). The flapping motion results from the fluttering instability of the plate s straight position uner the competing effects of soli inertia, aeroynamic pressure forcing an soli flexural rigiity. In that regar, the nature of the instability is similar to that of flow-conveying pipes (Païoussis, 99). Beyon the funamental interest of this instability an the resulting flapping motions, this mechanism is at the origin of the flapping of flags in the win an is also of interest for engineering (Watanabe et al., ) an biomeical applications (Huang, 99; Balint an Lucey, ). Hyroelastic instabilities of parallel-plate assemblies are also important in the stuy of inustrial cooling systems, as foun in some nuclear reactors (Miller, 9; Kim an Davis, 99; Guo an Païoussis, a). Several moels have been propose to unerstan the instability threshol in the case of a single elastic plate. Shelley et al. () consiere the one-imensional linear stability of a flag of infinite span an length uner parallel flow forcing. Lemaitre et al. () use a slener-boy approximation to stuy the linear stability of long elastic ribbons. Kornecki et al. (97) stuie the linear stability of a finite-length cantilevere plate of infinite span, using a representation of the vortical wake as a vortex sheet (Theoorsen, 93; Bisplinghoff et al., 9), an Argentina Corresponing author aress: smichelin@ucs.eu (Sébastien Michelin) Preprint submitte to Elsevier September, 9

2 an Mahaevan () use a similar approach to stuy the influence of in-plane tension an finite span on the stability threshol. Guo an Païoussis (b) consiere the stability of a finite-length two-imensional plate in a channel with various bounary conitions, using a simpler representation of the wake. Eloy et al. (7) use the same approach to investigate the influence of three-imensional effects on the stability of a finite-length cantilevere plate. In the case of multiple parallel elastic plates, the impose flow acts both as a forcing an a coupling mechanism between the motion of the plates. When the istance between the plates is of the orer of or less than their length, it is expecte that their ynamics will be strongly couple. Zhang et al. () generalize their soap-film experiment to two parallel filaments an showe that the filaments ynamics became couple an that, epening on their relative istance, inphase an out-of-phase moes coul be observe (Figure ). This observation was confirme by Jia et al. (7) who also consiere filaments of various lengths. Depening on the length of the filaments, they observe either a transition from out-of-phase to in-phase flapping when bringing the filaments close together, or a persistence of the out-of-phase moe even at small separation. These results suggest that the nature of the observe regime is strongly epenent on more than just the relative istance between the ifferent plates or filaments. Numerical simulations (Zhu an Peskin, 3; Farnell et al., ; Huang et al., 7; Michelin an Llewellyn Smith, 9) have also reporte the existence of these two classes of regimes. Tang an Païoussis (9) presente a numerical stuy of the stability criteria for the in-phase an out-of-phase moes. More recently, Schouveiler & Eloy (9, private communication) consiere three an more couple plates in a win-tunnel experiment an observe various moe structures (Figure ). In particular, in the symmetric moe, the mile plate was observe to remain still. Guo an Païoussis (a) stuie the linear stability of an infinite number of parallel plates of finite aspect ratio, clampe on their sie eges an free at the leaing an trailing eges. They however focuse only on one particular type of moes (the out-of-phase moe, where the motions of two consecutive plates are always opposite), using the resulting symmetries to significantly reuce the complexity of the system. The purpose of the present stuy is to propose a linear stability analysis of N couple flexible plates of infinite span, clampe at their leaing ege in a uniform impose flow. We investigate the existence an nature of couple moes, that were previously observe experimentally an numerically (Zhang et al., ; Zhu an Peskin, 3; Jia et al., 7), as well as the influence of the plates separation on the stability of the trivial state of rest an the structure of the linearly ominant flapping moes. In Section, the ouble-wake metho use in Guo an Païoussis (b) an Eloy et al. (7) for a single plate is extene to the case of an arbitrary number of equiistant parallel plates. In Sections 3 an, results on the linear stability of respectively two an three plates an the structure of the most unstable moes are iscusse. Section proposes an overview of expecte results at larger N as well as a iscussion of the influence of N on the critical stability curve for a fixe istance between the plates, an the nature of the corresponing ominant moes. Finally, conclusions are rawn in Section.. Aeroynamic coupling of N flexible plates.. Description of the problem We are intereste here in the linear stability of N horizontal ientical flexible an inextensible plates, place in a parallel impose flow along the horizontal irection. The plates span is assume to be infinite so that the flow an plates isplacements can be consiere two-imensional. The plates are clampe at their leaing ege an are equally space along the vertical axis at a istance

3 In phase moe of two plates Out of phase moe of two plates U U In phase moe of three plates Out of phase moe of three plates Symmetric moe of three plates U U U Figure : (Top) Sketch of the in-phase an out-of-phase moes observe experimentally for two plates (Zhang et al., ; Jia et al., 7). (Bottom) Sketch of the in-phase, out-of-phase an symmetric moes observe experimentally for three plates (Schouveiler & Eloy, 9, private communication). D from each other: the leaing ege of plate j ( j N) is locate at (,(j )D). All plates have length L in the streamwise irection, mass per unit area ρ s an flexural rigiity B = Eh 3 /[( ν )] (per unit length in the thir imension). Here, E an ν are respectively the Young moulus an Poisson ratio of the plates, an h is the plate s thickness (h L). The velocity an ensity of the impose flow are ρ an U. The problem is characterize by three nonimensional parameters (M,U,) efine as M = ρl ρ s, U ρs = U L B, = D L. () In the following, all quantities are nonimensionalize using ρ, U an L as reference values: positions are nonimensionalize by L, time is nonimensionalize by L/U, velocity by U an pressure by ρu. We are intereste in the linear stability of the state of rest for which all the plates are horizontal an parallel to the incoming flow. Assuming that the isplacement of each plate from its state of rest is small, the geometry of plate j is escribe by the vertical isplacement y (j) (x,t) of the point locate at a istance x along plate j in the state of rest. We are intereste in the stability of infinitesimal perturbations, therefore y (j). We assume that the plate is inextensible: any plate element must conserve its length at all time. Therefore, if ξ (j) (x,t) is the horizontal isplacement of the point locate at a istance x along plate j in the state of rest, the inextensibility of the plate can be expresse as ( + ξ(j) x ) + ( 3 ) y (j) =. () x

4 (N ) (N ) Domain N Domain N (N) (N ) y (j ) U [p] (j) = (j) (j) (j ) (j) φ =φy =y +y (j) y t x [p] (j) = Domain Domain Domain (3) () () x Figure : Horizontal flow over N parallel plates. The plates inex (j) is inicate as well as the N + omains in wich the flui is partitione. On each plate, the continuity of the normal velocity is enforce. Continuity of pressure is enforce upstream an ownstream along the irection of each plate. This relation implies that if y (j) is of orer ε (with ε ), then the horizontal isplacement ξ (j) (x,t) is of orer ε, an we neglect this horizontal isplacement here. Assuming that the ynamics is governe by the Euler Bernoulli equation with clampe-free bounary conitions, an keeping only the ominant terms, the equation of motion of plate j is obtaine as: y (j) t = U y (j) x M [p] (j) (x), y (j) (,t) = y (j) x (,t) = y (j) xx (,t) = y (j) xxx(,t) =, (3) where [p] (j) is the pressure ifference between the upper an lower sies of plate j... Potential flow an wake representation The flui omain is ivie into N + omains (see Figure ): Domain : y <, Domain j ( j N ): (j ) < y < j, Domain N: y > N. We assume that the flow is potential in all omains, an the flow potential is continuously ifferentiable in each omain. In (3), we have [p] (j) (x) = p (j) (x,(j ) + ) p (j ) (x,(j ) ). From the Bernoulli equation, the pressure in omain j is given by p (j) (x,y) = Φ(j) t Φ(j), ()

5 where Φ (j) is the total flow potential, which can be ecompose as Φ (j) (x,y) = x + φ (j) (x,y), () with φ (j) the perturbation of the potential ue to the motion of the plates away from their state of rest ( φ (j) ). Keeping only the linear terms, the pressure forcing on each plate becomes ( [p] (j) (x) = t + ) (φ (j) φ (j ) ), () x evaluate at y = (j ). The major ifficulty in such situations is the representation of the wake. In the case of one single flapping plate, several moels have been propose. Using Theoorsen s theory (Theoorsen, 93; Bisplinghoff et al., 9), Kornecki et al. (97) represente the wake of a single flexible plate as a istribution of vorticity along the horizontal axis avecte at leaing orer by the impose flow. The intensity of the vorticity she at each time was such that the regularity conition is enforce at the trailing ege. In the following, we refer to this metho as the Vortex Sheet (VS) metho. Computations with better accuracy using the same metho an assumptions were reporte recently (Eloy et al., ). The Vortex Sheet metho was recently compare successfully to experimental an numerical results (Eloy et al., ; Michelin et al., ). This elegant metho is, however, ifficult to generalize to two plates or one plate close to a wall, as it is base on the possibility of isolating the contribution of the frequency to the circulatory pressure with the so-calle Theoorsen s function (Bisplinghoff et al., 9). An alternative approach has been propose, representing the presence of the wake by the continuity of the pressure across the horizontal axis ownstream from the plate ([p] (j) (x) = for x > ); Guo an Païoussis (b) an Eloy et al. (7) use this metho to stuy, respectively, the two-imensional stability of a plate with infinite span in an invisci channel flow an the three-imensional stability of a finite-span plate in an unboune flow. In this metho, the pressure istribution is solve in Fourier space an the absence of singularities at both eges is assume, thereby introucing an upstream wake to eal with the leaing ege singularity. In the following, we shall therefore refer to this metho as the Double Wake (DW) approach. Although this ouble-wake assumption is not base on any physical argument, it was observe that in the case of a single plate, it leas to a goo preiction of the critical velocity for flutter instability (Eloy et al., 7; Watanabe et al., ), particularly for lighter plates (M ). Figure 3 shows that the velocity threshol obtaine using the DW metho is in goo agreement with the preiction of the VS metho for M. Both methos (VS an DW) also preict the existence of ifferent branches in the stability curve, each branch corresponing to ifferent moe structures being most unstable. The first branch correspons to a moe (one-neck moe), the secon branch to a moe 3 (two-neck moe), an so on. Figure 3(b) shows the superposition of the stability curves obtaine with each metho, an the maximum growth rate observe using the DW representation. This provies some insight on the behavior at small M. The VS moel preicts a stable flat state while the DW preicts unstable behavior. However, the associate growth rates (obtaine with the DW metho) are very small in the region of isagreement between the two theories. The ifference between the two approaches resies solely in the treatment of the wake. At high values of M, the ominant moes have shorter wavelengths an it is therefore expecte that their ynamics are less influence by the wake an more by the local isplacement of the plate.

6 (a). (b).. Figure 3: (a) Critical stability curve of a single cantilevere flexible plate of infinite span using the ouble wake (DW) representation (soli) an using the vortex sheet (VS) wake moel (ashe). (b) Stability curves of (a) superimpose on the maximum growth rate observe with the ouble wake representation. Conversely, the first branch correspons to moe with a characteristic spatial scale of the orer of the length of the plate, an is strongly influence by the escription of the wake..3. Computation of the pressure forcing We follow the DW metho to compute the pressure forcing on each plate (Guo an Païoussis, b; Eloy et al., 7). Each plate s isplacement is ecompose into normal moes y (j) (x,t) = Re [ ζ (j) (x)e iωt] with ω a priori complex. Re(ω) an Im(ω) are respectively the moe s frequency an growth rate. Unstable moes correspon to positive growth rate (Im(ω) < ) an stable moes to negative growth rate (Im(ω) > ). All quantities (pressure, potential) are also proportional to e iωt, an in the following the time-epenence e iωt will be implicit. The flow potential must satisfy the following system: φ (j ) y φ (j) = φ (j) x + φ (j) y =, j N, (7a) φ () y = φ (N) y =, (7b) (x,y = (j )) = φ(j) y (x,y = (j )) = iωζ(j) + ζ(j) x, for x, (7c) [p] (j) (x) =, for x an x, (7) with the pressure jump obtaine from Bernoulli s equation () as ( [p] (j) (x) = iω + x) [ ] φ (j) (x,(j )) φ (j ) (x,(j )). () Equation (7b) prescribes the correct ecay of the flow perturbation: far from the plates in omain () an (N), the velocity is ominate by the impose unit parallel flow, an the perturbation velocity becomes negligible. Conition (7c) enforces the slip bounary conition on each plate: on each sie of the plate, the normal velocity of the flow must be equal to the plate s normal velocity. Eq. (7) is the regularity conition introuce by the wake ownstream of the plates an the

7 fictitious upstream wake: from (), we see that it is equivalent to the avection by the mean flow of a iscontinuity in the flow potential, or equivalently of a vorticity istribution on the horizontal axis (Eloy et al., 7). Using the Fourier transform in x, we look for solutions of the form φ (j) (x,y) = an from (7a) the function φ (j) (k,y) must satisfy whose general solution can be written From (7c), we obtain: φ (j) (k,y)e ikx k, (9) φ(j) y k φ(j) =, () φ (j) (k,y) = A (j) (k)e k y + B (j) (k)e k y. () B (j) (k) A (j) (k)e k (j ) = B (j ) (k) A (j ) (k)e k (j ), for j N. () The pressure ifference at y = (j ) is then compute from () an () as [ ] [p] (j) (x) = i (ω + k) A (j) (k) A (j ) (k) e k (j ) e ikx k = i [ ] (ω + k) B (j) (k) B (j ) (k) e k (j ) e ikx k. Differentiating (3) with respect to x an using the inverse Fourier transform an (7), we obtain A (j) (k) A (j ) (k) = B (j) (k) B (j ) (k) = e k (j ) πk(ω + k) e k (j ) πk(ω + k) [p] (j) (ξ)e ikξ ξ, [p] (j) (ξ)e ikξ ξ, (3a) (3b) (a) (b) where the prime enotes a erivative with respect to x ([p] (j) is therefore the horizontal graient of the pressure jump on plate j). Using (7b), A () = B (N) = an therefore A (j) (k) = B (j) (k) = πk(ω + k) πk(ω + k) j e k (l ) [p] (l) (ξ)e ikξ ξ, l= N l=j+ e k (l ) [p] (l) (ξ)e ikξ ξ. Finally, applying the operator (iω + /x) to (7c), an using () an (), ω ζ (j) + iω ζ(j) x + ζ (j) x = i = i π = π N l= k (ω + k) sgn(k) 7 ( B (j) e k (j ) A (j) e k (j )) e ikx k N e k(j l) [p] (l) (ξ)e ik(x ξ) ξk. l= (a) (b) K j,l (x ξ)[p] (l) (ξ)ξ ()

8 with K j,l (u) = i Using the following pair of Fourier transforms we obtain sgn(k)e k(j l) e iku k. (7) i sgn(k) u, e k α α α + u, () K j,l (u) = u u + (j l) for {j,l} N. (9) We therefore have to solve the following system of integral equations for the pressure graients along each plate [p] (j) : [p] (j) (ξ)ξ + π x ξ π l j x ξ (x ξ) + (j l) [p] (l) (ξ)ξ = ω ζ (j) + iω ζ(j) x + ζ (j) x, () for j N, with the aitional constraints that [p] (j) = at x = {,} so that [p] (j) x =. In (), we have purposely isolate the term j = l which gives a Cauchy-type singular kernel. The first integral is unerstoo as the Cauchy principal value. The linearity an form of () suggests to look for a solution of this system of integral equations in the following form (see for example Eloy et al., 7): [p] (j) (x) = [p (K) ] (j) (x) + iω[p (G) ] (j) (x) ω [p (M) ] (j) (x), () with x an j N. The pressure forcings [p (K) ] (j), [p (G) ] (j) an [p (M) ] (j) are solutions of [p (K) ] (j) (ξ)ξ + x ξ π x ξ π (x ξ) + (j l) [p(k) ] (l) (ξ)ξ = ζ (j) x [p (G) ] (j) (ξ)ξ π x ξ [p (M) ] (j) (ξ)ξ π x ξ with + π with + π with l j l j l j [p (K) ] (j) x =, (a) x ξ (x ξ) + (j l) [p(g) ] (l) (ξ)ξ = ζ(j) x [p (G) ] (j) x =, (b) x ξ (x ξ) + (j l) [p(m) ] (l) (ξ)ξ = ζ (j) [p (M) ] (j) x =. (c) In () an (), the pressure contributions with superscripts K, G an M correspon respectively to ae rigiity, gyroscopic an ae inertia effects (Païoussis, ).

9 .. Galerkin ecomposition The stability problem is then investigate using the Galerkin metho. The isplacement of each plate is ecompose onto the Q first normal moes of the clampe-free beam in vacuum ζ (j) (x) = Q n= α (j) n ψ n (x), ψ n (x) = cosh(λ n x) cos(λ n x)+ sin(λ n) sinh(λ n ) cos(λ n ) + cosh(λ n ) [sinh(λ nx) sin(λ n x)], an λ n are the successive positive roots of These moes are solutions of the following problem: (3) + cos λ n cosh λ n =. () ψ n x = λ nψ n, ψ n () = ψ n() = ψ n() = ψ n () =. () From this basis for one plate, we can easily buil the basis for the N-plate problem by consiering the moes Ψ m ( m N Q) efine as follows: moe (i )Q + q (with q Q an i N) correspons to a isplacement equal to ψ q for plate i, all the other plates being hel fixe. (In other wors, if ψ is the matrix of the value of the moes (3) at a set of points x, the basis Ψ for the N-plate problem is block-iagonal with N iagonal blocks equal to ψ.) From (), for each moe Ψ m, we obtain the pressure graient P m as the following superposition: P m = P (K) m [ with P m (K) = [p (K) ] () m,[p (K) ] () m,...,[p (K) ] (N) m π [p (K) ] (j) m (ξ)ξ + x ξ π l j + iωp (G) m ] T an from (), ω P (M) m, () x ξ (x ξ) + (j l) [p(k) ] (l) m (ξ)ξ = Ψ (j) m x, [p (K) ] (j) m (x)x = (7) with Ψ (j) (i )Q+q (x) = δ ijψ q (x). P (G) m an P (M) m are obtaine in the same way from (), an the ifferent pressure fiels are then recovere by integration from the leaing ege. For the N plates, Eq. (3) becomes (noting that most of the coefficients of each Ψ m are zero): N Q m= [ α m ω Ψ m + Ψ ( m U x + M ω P m (M) + iωp m (G) ) ] + P m (K) =. () We efine the scalar prouct. of F = [ f () (x),..,f (N) (x) ] T an G = [ g () (x),...,g (N) (x) ] T as F,G = N j= f (j) (x)g (j) (x)x. (9) Taking the scalar prouct of () with Ψ n, we obtain a nonlinear eigenvalue problem in ω: [ ω (M + M F (M) ) + iωm F (G) + ] U K + M F (K) α = (3) 9

10 where F (M) mn M mn = Ψ m,ψ n, K mn = (M) = P m,ψ n, F mn (G) = P (G) m,ψ n, Ψ m x,ψ n F (K) mn (K) = P m,ψ n. (3) Because Ψ (j) (i )Q+q = δ ijψ p (x), an the natural moes of the clampe-free beam are orthogonal an satisfy (), M is the ientity matrix an K is iagonal with with λ q efine in ()... Numerical solution K (i )Q+q = λ q for i N an q Q, (3) The main task consists in etermining the matrices F (M), F (G) an F (K). The results presente thereafter were obtaine using Q = moes for each plate. A large enough number of Galerkin moes must be use to properly resolve the structure, growth rate an frequency of the physical moes [see for example Lemaitre et al. ()], an tests with Q = an Q = 9 were performe to ensure a proper convergence. Equation (3) has N tot = N Q eigenmoes an the moes with largest frequency arise from the Galerkin ecomposition an are not physical. We therefore chose to retain only the N tot / moes with smallest frequency in the remainer of the paper (N moes for Q = ) to ensure we are stuying only physical moes. This choice might lea us to iscar some physical moes, but such iscare moes woul have a very high frequency an short wavelength. Therefore, we expect these moes to have a negligible effect in the analysis of the real problem, as flow viscosity an material amping (both neglecte in the present potential flow approach) woul lea to a rapi issipation of the energy in those moes. Each function ψ n (x) is evaluate on N p = Gauss-Chebyshev points, an expane using the first N p Chebyshev polynomials of the first kin. Using this iscretization is particularly well-suite for the solution of the couple singular integral equations (). The proceure for solving () is etaile in Appenix A. Once the ifferent [p (K) ] (j), [p (G) ] (j) an [p (M) ] (j) have been obtaine, the matrix coefficients are compute using (9) an a Gauss Chebyshev quarature (Pozrikiis, 99). Once the matrices F (M), F (G) an F (K) have been obtaine for a given value of, the nonlinear eigenvalue problem (3) is solve for given values of M an U. We first note that, in (3), ω always appears as powers of iω an all matrices are real. Therefore, if ω is a solution, ω is as well, with the same moe structure characteristics (the eigenfunctions are conjugate pairs). This result is a irect consequence of the choice mae for the moal ecomposition: y (j) (x,t) = Re [ ζ (j) (x)e iωt]. The eigenvalues are therefore of two kins: purely imaginary an pairs of eigenvalues with the same imaginary parts but opposite real parts. Because of this structure, in the following we will only iscuss the eigenvalue with positive real part, keeping in min that when this real part is strictly positive, an associate eigenvalue also exists with opposite real part. In the following, we stuy successively the stability properties of N =, N = 3 an N parallel elastic plates. In several places, we will consier the most unstable moe to etermine for example the ominant moe s nature. We must point out here that this moe is ominant in the linear regime. Although several numerical an experimental stuies on flexible flags have reporte

11 Figure : Critical stability curve of the state of rest of two plates in the (M, U )-plane for =. (circle), =. (thin soli), =. (ashe), = (otte) an = 3 (ash-otte). The thick soli line is the stability curve for one flag only using the same approach (Eloy et al., 7) an is equivalent to =. a goo agreement between the structure of the observe ominant moe an the preiction of linear stability analysis [e.g Eloy et al. (), Michelin et al. ()], in the fully-eveloppe regime, nonlinear phenomena such as moe competition coul moify the nature or properties of the ominant moe (e.g. frequency). This will therefore limit our ability to perform a quantitative comparison of our linear stability results to the experimental work of Zhang et al. () an Jia et al. (7), as the motion of the filaments observe in these experiments correspon to a fully-eveloppe nonlinear regime. Furthermore, although soap-film experiments represent a goo approximation of two-imensional flows, this approximation is only vali far from the filaments: the wetting of the filaments creates a bulge of flui whose influence on the inertia an rigiity of the filament can only be estimate roughly [see Zhu an Peskin (3)]. A qualitative comparison of the moe nature an properties with the experimental observations can however be performe. 3. Stability of two couple plates In this section, we focus on the case of two ientical flexible plates place at a relative istance from each other. The metho outline in the previous section is use setting N =. 3.. Influence of on the stability of the flat state of rest One question of interest when approaching two flexible plates to each other is whether an how the presence of the other plate will influence the stability threshol of the state of rest (flat an parallel to the impose flow). The critical stability curve is plotte on Figure for various values of the relative istance between the plates leaing eges. Below the critical curve, the state of rest is a stable equilibrium; above it, this state of rest is unstable an flapping can evelop. As a general result, we observe that the hyroynamic coupling between the plates tens to estabilize the state of rest, as the stable range of parameter values (M,U ) shrinks with

12 ecreasing. We observe that this reuction of the stability threshol U c (M,) with (or the minimum U above which the state of rest becomes unstable) is not uniform with respect to M. Heavy flags (M ) are estabilize for istances between the plates as large as three times their lengths, while the stability of light plates (M ) is not affecte until <.. This is confirme by looking at the growth rate of the most unstable moe, plotte on Figure (a) for several values of the relative istance between the plates. Two effects are evient from this picture. As expecte for a particular istance, the growth rate increases above the critical stability curve for increasing U : increasing U correspons to an increasing forcing flow or a relatively smaller rigiity of the plates. More kinetic energy is then transferre from the outsie flow to the plates motion. For small, the ominant moe s stability properties are moifie the most for small M : in that case, a significant increase of the ominant moe s growth rate is observe for ecreasing. The comparison of the ifferent pictures in Figure (a) also shows the evolution with of the location of branches with reuce instability (light colors) in the (M,U )-plane, information that is not available when only the critical stability curve is consiere. These regions are the most susceptible to become stable if a small amount of issipation is ae to the system. This ifference of behavior between small M an large M is illustrate on Figure where the ominant moe characteristics are stuie in the (U,)-plane. For M =, the ominant moe stability, nature, growth rate an frequency epen strongly on, while for M =, has a much weaker influence an the frequency an growth rate variations are ue to changes in the ominant moe rather than a moification of a particular moe s properties with. One interpretation of the preferential estabilization of flags with low M lies in the moe structure associate with each branch on the stability curves in Figures 3 an. The most unstable branch at low M correspons to moes with longer wavelengths than the subsequent branches. Moes with shorter wavelengths are expecte to be less influence by the presence of the secon plate as the relevant aspect ratio is λ/d rather than L/D for this particular moe, with λ the wavelength of the moe. 3.. In-phase an out-of-phase moes For two couple plates, eigenmoes can be of two natures: in-phase (the two plates having the same motion) an out-of-phase (the two plates having symmetric motions with respect to the horizontal axis) as illustrate on Figures an 7. The nature of the moe is etermine by computing the correlation between the spatial structure of the moe component along each plate. For a particular moe ω, each plate s isplacement can be efine as y (j) (x,t) = Re[ζ (j) (x)e iωt ] with the same complex frequency ω. The correlation coefficient for the motion of plates m an n is efine as the average value in x of the time correlation of these plates isplacement at x: [ ] Re ζ (m) (x)ζ (n) (x) r mn = ζ (m) (x)ζ (n) x, (33) (x) where the integran is just the relative phase between ζ (m) (x) an ζ (n) (x). This efinition remains vali for the case of multiple plates. Here there are only two plates an only one correlation r = r. In practice, within each moe of the two-plate system, the isplacements of the plates were foun either equal or opposite an a symmetry argument for this is given in Section. Therefore, the correlation coefficient r is equal to ±. A moe with r = (resp. r = ) is calle thereafter an in-phase (resp. out-of-phase) moe. Figure (b) shows for given values of, the nature of the most

13 (a) (b) Figure : (a) Growth rate an (b) nature of the most unstable moe plotte in the (M, U )-plane for (from top to bottom) =., =., = an = 3. On the right column, the stability region of the state of rest is shown in black, while grey (resp. white) regions correspon to an out-of-phase (resp. in-phase) ominant moe. 3

14 Figure : Evolution of the ominant moe properties in the (,U )-plane for M = (left) an M = (right). (Top) Regime observe (with the same color coe as in Figure ). (Center) Growth rate of the most unstable moe (stable regions are inicate with a zero growth rate). (Bottom) Frequency of the ominant moe.

15 ..... =. Moe I x..... =. Moe O..... x Figure 7: Dominant moe for M = an U = 3 for (left) =. an (right) =.. The position of each plate is plotte at t = t (soli), t = t + T/ (ashe), t = t + T/ (ash-otte) an t = t +3T/ (otte), with T the pseuo-perio of the moe (π/re(ω)). The same arbitrary amplitue of vertical motion was chosen for both cases. (a) (b) Moe frequency 7 3 I O O I3 O3 O I I Ia+b Oa+b Growth rate 3 I3 O I O3 I I O O Ia Oa Ib Ob Figure : M = an U = 3. Variations of (a) the frequency an (b) the growth rate of the four in-phase (thick lines) an out-of-phase (thin lines) least stable moes. The moes are numbere by increasing frequency in the limit. unstable moe in the (M,U )-plane. Alternatively, Figure shows the evolution of the nature of the ominant moe in the (,U )-plane for given M, thereby giving some insight on the influence of. In the following, moes will be reference by their nature (I for in-phase moes an O for outof-phase moes) an numbere by increasing frequencies at large, to correspon for convenience to the numbering of the one-plate case (it is therefore possible for a moe to have a higher frequency than a moe 3 at shorter istances). These maps show that for increasing, the omain of stability of the flat state of rest increases, as observe in the previous section. From Figures an, we also observe that for given values of M an U, it is possible to observe an in-phase moe at small istances an a transition to an out-of-phase regime for increasing plate separation (see for example the case M = an U = 3 in Figure an in more etail in Figure ). This transition from in-phase to out-of-phase moes with increasing separation was observe experimentally (Zhang et al., ; Jia et al., 7). While it is clear from Figure (b) that the in-phase flapping moes are more often ominant at

16 Moe frequency (a) O I O I3 O3 I Oa+b Ia+b I Oa Ia O Growth rate 3 (b) O3 O I3 I Ia Oa+b Oa Ib Ob O I O Figure 9: Same as Figure for M = an U =. small istances than at larger istances, this transition is not always observe: in-phase flapping can be observe for large (up to five times the plate s length for M = an U =. as seen on Figure ) an out-of-phase flapping can be observe for all values of (for small inertia ratio M, the ominant moe is always out-of-phase). This is consistent with the soap-film experiments reporte in Jia et al. (7), which observe an out-of-phase moe at small an large istances for the shortest filament length (corresponing to smaller M an U, if all other properties are kept constant). It is also consistent with the results of Tang an Païoussis (9), who observe that the velocity threshol of the fluttering instability was always lower for out-of-phase moes than for in-phase moes in their stuy of plates with M = Influence of on the moe properties Figures an 9 show the evolution with of the frequency Re(ω) an growthrate Im( ω) of the lowest frequency in-phase an out-of-phase moes for two ifferent sets of values of (M,U ). The branches on each of these plots correspon to continuous variations of the eigenvalue ω. Note also that the nature of the moe (in-phase/out-of-phase) remains unchange on a particular branch for all -values (the correlation coefficient for each branch remains very close to either or ). The structure of the moe (e.g. number of necks) can however vary with, although we note that the most unstable moes preserve their general structure throughout the -omain (Figure ). Other more ampe moes can however experience a strong moification of their structure as can be seen for moe Ob for example on Figure. The existence of a zero in the envelope correspons to a staning wave moe (Re(ω) = ). At large istances, we observe the pairing of in-phase an out-of-phase branches. Within each pair, the moe frequencies, growth rates an moe structures become ientical an approach asymptotically the moe characteristics obtaine in the case of a single plate (Figures an ). Within each pair, the two moes iffer however by the values of the correlation coefficient r (or the phase ifference between the plates motion) which are opposite. We showe that the variations of the moes growth rate with, M an U can inuce a suen change of the ominant moe nature as one parameter is varie (corresponing to the ifferent omain bounaries on Figures (b) an ). If the most linearly unstable moe etermines the properties of the observe flapping in experiments an numerical simulations, then by varying one of the parameters (by either changing the separation D or the flow velocity U for example),

17 Moe enveloppe... =. =. =. = = Moe Ia Moe enveloppe... =. =. =. = = Moe Ib Moe enveloppe... =. =. =. = = Moe I x.... x.... x Moe enveloppe... =. =. =. = = Moe Oa Moe enveloppe... =. =. =. = = Moe Ob Moe enveloppe... =. =. =. = = Moe O x.... x.... x Moe enveloppe... =. =. =. = = Moe I3 Moe enveloppe... =. =. =. = = Moe I x.... x Moe enveloppe... =. =. =. = = Moe O3 Moe enveloppe... =. =. =. = = Moe O x.... x Figure : Evolution with of the envelope of the least stable in-phase an out-of-phase moes for M = an U = 3. The inexing of the moes correspon to Figure. The envelope is compute as the maximum vertical eflection achieve for x, for one of the two flags after normalizing the moe amplitue so that the amplitue at the trailing ege is always unity. 7

18 one will experience a suen change in the observe properties (for example, the flapping moe shape or its frequency). For M = an U = 3, the ominant moe is an in-phase flapping at short istances ( <.) an an out-of-phase flapping at larger istances (Figure ). At the transition, the structure of the ominant moe (an therefore its characteristic wavelength) is also moifie from a -neck moe (I3) to a 3-neck moe (O) (see Figure 7), an a frequency jump is observe from ω =. (moe I3) to ω =.9 (moe I). This frequency switch was also reporte in previous experimental an numerical stuies (Zhang et al., ; Farnell et al., ; Jia et al., 7). The frequency jump can be observe even when a transition occurs between two moes of the same nature. Figure 9 illustrates this with a case where an out-of-phase moe is always ominant for M = an U =. Moe O is ominant for small istances, while O3 is ominant at larger istances. The moe switch at.3 leas to a negative ominant frequency switch from ω =. to ω = 3.7. This frequency jump an its variability with the parameter values are clearly visible on Figure. The frequency plots show some large variations of the ominant frequency with M an U for given. Large frequency values can be achieve at low U (for large value of the plates rigiity). We must point out that these frequencies occur mostly in the stability region where all moes have negative growth rates. In some situations, large ominant frequencies can also occur for unstable configurations but at very small separation istance (typically less than % of the plates length as seen for M = on Figure ). In the presence of a small amount of material amping or flow viscosity (neglecte here), we expect these high-frequency moes to be the most affecte.. Stability of three couple plates In the case of three evenly-space ientical plates, three ifferent types of moes are obtaine that correspon to the moe types observe experimentally (see Figure ): (i) an in-phase moe (I) where all three plates flap in phase; the correlation coefficients as efine in (33) satisfy r = r 3 = r 3 = an the outer plates have the same motion; (ii) an out-of phase moe (O) where the outer plates flap in phase with the same amplitue an the mile plate flaps with a phase equal to π with respect to the other plates (r = r 3 = an r 3 = ); (iii) a symmetric moe (S) where the outer plates have opposite vertical isplacements an the mile plate remains still: r 3 = an ζ () (x) =. A symmetry argument for the existence of moe (S) will be given in Section. The same symmetry argument will also prove that plates an 3 always have equal or opposite motions, therefore their relative motion amplitue is equal to. However, in moes I an O, the motion of the center plate is in general not of the same amplitue as the two others (although it was observe that it ha the same x epenence leaing to a correlation coefficient of ±). For a given value of M, Figure shows the evolution of the moe properties in the (,U )- plane. We observe that at short istances an for light plates (large M ), the ominant moe structure is highly-sensitive to the problem parameters, with the presence of finger-shape regions of ominance of the (O) an (S) moes. At large istances the out-of-phase moe (O) prevails as for two plates. However for large, the moes group in triplets. Within each triplet, one fins a moe of each kin (I, O an S) with frequency, growth rate an envelope asymptotically equal to the values for one plate only (Figure ). For the particular choice M = 3 an U = 3, we

19 observe that for increasing, the ominant moe takes alternatively each of the three possible natures (Figure ). As for two plates, a switch in the ominant moe can lea to a suen change of all the following: moe nature (I,O,S), moe structure an frequency.. Results for larger N.. Symmetry argument In the previous sections, we have observe that all the moes are always symmetric or antisymmetric in structure with respect to the axis of symmetry (between the two plates for N = or on the mile plate for N = 3). We attempt here to provie a physical argument for this observation. A mathematical proof involving the structure of the matrices in (3) is presente in Appenix B. The orientation of the horizontal axis is physically etermine by the forcing flow. The orientation of the vertical axis is however purely arbitrary. For a particular moe, efine ζ (j) (x) the isplacement of the plate clampe at y = (j ). Defining a new y-coorinate y as y = (N ) y an a new inexing of the plates j = N + j will leave the problem absolutely unchange. In this new situation, the isplacement of the plate sitting at y = (j ) in a particular moe will be ζ (N+ j) (x). However, the physical problem is unchange an the moe itself cannot have been moifie, therefore there must exist a constant λ such that ζ (j) (x) = λζ (N+ j) (x) for all j N. (3) Consiering the previous equality for any pair j = p an j = N + p ( p N), we see that λ = ±. In a particular moe, the isplacements of plates j an N + j must therefore be equal or opposite. The correlation coefficient of the two plates as efine in (33) must be equal to ±. Therefore, the motion of the N plates must be symmetric or anti-symmetric with respect to the plane locate between plates N/ an N/ + (resp. on plate (N + )/) for even N (resp. o N). One consequence of this result is that for o values of N, symmetric moes correspon to a center plate at rest, as was observe for the case of three plates in Section... The limit N We now consier the limit case of a large number of plates N. This geometry is particularly relevant in the stuy of plate assemblies as foun in cooling systems or in some nuclear reactors (Miller, 9; Kim an Davis, 99; Guo an Païoussis, a). Since the problem is invariant by a translation of along the vertical axis, we look for solutions of (7) in which the isplacement ζ (n) an pressure perturbation [p] (n) of plate n ( < n < ) are of the form ζ (n) (x) = ζ () (x)e inϕ, [p] (n) (x) = [p] () (x)e inϕ, (3) with the phase ϕ being arbitrary between an π. From (), we obtain [p] () from ζ () as π K (x ξ,ϕ)[p] () (ξ)ξ = ω ζ () + iω ζ() x + ζ () x, with the moifie kernel K given by K (u,ϕ) = u + 9 n= [p] () (ξ)ξ = (3) ucos nϕ u + n. (37)

20 * U * U U* U *. U *. * U.... Figure : Evolution of the ominant moe properties for three plates with M = 3 (left) an M = (right). (Top) The nature of the ominant moe is given in the (, U )-plane: stability region (black), symmetric moe S (ark gray), out-of-phase moe O (light gray) an in-phase moe I (white). (Center) The maximum growth rate is plotte in the (, U )-plane. Stability regions correspon to a zero growth rate for clarity. (Bottom) Contours of the ominant moe s frequency in the (, U )-plane.

21 (a) 7 I (b) O I3 I3+O3+S3 Moe frequency 3 S O S3 S I O I O O3 I3 S I+O+S I3+O3+S3 I+O+S I+O+S Moe growth rate 3 S3 S I O3 Sa I I Oa S O Sb Ob I+O+S I+O+S I+O+S Figure : Evolution of the frequency (left) an growth rate (right) of the first four in-phase, out-of phase an symmetric moes for three plates with M = 3 an U = 3. The branches corresponing to in-phase, out-of-phase an symmetric moes are plotte respectively with thick soli lines, thin soli lines an otte lines. We note here that K is an even function of ϕ (as expecte since the problem is invariant by reversing the labeling orer of the plates). In the following, we therefore restrict our stuy to ϕ π. Using Fourier series, we can etermine K explicitly as K (u,ϕ) = π cosh [ u (π ϕ)] sinh ( ) πu, for ϕ π. (3) The case ϕ = will correspon to a purely in-phase moe (all the plates will have the same motion) an the case ϕ = π correspons to the purely out-of-phase moe (any two consecutive plates have opposite vertical isplacements). Guo an Païoussis (a) stuie the latter moe exclusively by assuming the symmetry property to simplify the potential flow problem. They also use a ifferent geometry with free bounary conitions applie at the leaing an trailing eges, the sie eges of the plate being clampe to a rigi wall. Here, we o not assume a priori a particular phase ifference ϕ but instea consier all possible values of ϕ to etermine the most unstable moe an its nature.... Stability threshol at large N For a given ϕ, we can use the same approach as in Section an ecompose ζ () over the first Q funamental moes of the clampe-free beam. For each moe ζ m (), we then obtain from (3) an () the pressure graients [p (M) ] () m, [p (G) ] () m an [p (K) ] () m. For each value of ϕ, we compute the critical stability curve in the (M,U ) plane of the moes with a phase ifference ϕ between successive plates, an obtain the absolute stability curve by taking the intersection of the stability omains. The results are shown on Figure 3 for two values of the relative istance of the plates =. an =., an compare to the results obtaine using the metho of section with an increasing number of plates. We observe that the results obtaine for increasing values of N converge well to this limit bounary as N. We also observe that the stability curve converges rather fast towar this limit for both values of. Most of the limit stability curve is obtaine with a very goo approximation for N.

22 Figure 3: Evolution of the critical stability curve in the (M, U )-plane for =. (left) an =. (right), an an increasing number of plates: N = (stars), N = (circles) an N = (triangles). The limit N escribe in section. is plotte as a thick black line.... Properties of the ominant moe in the N limit Using the same metho, we can compute the frequency an growth rate of the moes with a phase ifference equal to ϕ. For a given value of M, U an, we search over all the possible values of ϕ for the most unstable moe. The properties of the ominant moe are shown on Figure : the phase ifference between two consecutive plates in that moe, as well as this moe s frequency an growth rate, are successively stuie. While the out-of-phase moe (ϕ = π) tens to ominate for lower values of M an larger values of, it is not necessarily the case. In particular, for large values of M or small values of, the in-phase moe can be ominant. We also note that the transition from ϕ = to ϕ = π in the ominant moe is not necessarily iscontinuous but that ominant moes with any value of ϕ can be foun for a particular value of M, U or. This seems to contraict the assumption of Guo an Païoussis (a) that the out-of-phase moe is ominant in the linear regime. We must, however, point out that Guo an Païoussis (a) consiere free-free bounary conitions, while the present work consiers clampe-free plates. The change in bounary conitions is expecte to have a significant impact on the position of the critical stability curve in the (M,U )-plane (Guo an Païoussis, b). We note previously that the convergence of the stability curve with increasing N is a rather fast but inhomogeneous process; some parts of the curve are obtaine with a very goo approximation for N as low or, while others require a larger number of plates (Figure 3). From the top panel of Figure, we observe that the zones of fast convergence correspon to regions of the parameter space where the purely out-of-phase moe is ominant (ϕ = π), while the slowest converging regions correspon to smaller values of ϕ. Physically, a moe with a phase ifference ϕ between two successive plates can be seen as a wave traveling along the vertical axis with a wavelength equal to π/ϕ an the number of plates necessary to escribe that moe is of the orer of π/ϕ. For ϕ π, the instability is well escribe by the motion of a plate couple to its two nearest neighbors, while for small ϕ, a large number of plates is necessary to escribe the structure of the moe accurately. The stability an properties of local moes (those escribe by a small number of plates) are less influence by the total number of plates in the system.

23 Figure : Evolution in the (M, U )-plane of the properties of the ominant moe for an infinite number of equallyspace plates positione at a istance =. (left) an =. (right) from each other. (Top) Relative phase ϕ between two consecutive plates (normalize by π). ϕ/π = correspon to the purely out-of-phase moe where one plate s motion is opposite to its two neighbors. ϕ/π = correspons to the in-phase moe where all plates flap in phase. For reference, the stability curve of Figure 3 is also plotte (thick black line). (Center) Growth rate of the ominant moe. (Bottom) Frequency of the ominant moe. 3

24 . Conclusions Using the ouble-wake approach, we have performe the linear stability analysis of an array of N parallel an ientical flexible plates clampe at their leaing ege in a horizontal flow. The behavior of two an three plates was iscusse in more etail, as well as the limit of an infinite number of plates. This metho allows one to reprouce important properties of the couple motion of two plates observe in numerical an experimental stuies. For a finite relative istance between the plates, two types of moes are observe, corresponing to in-phase or out-of-phase motions of the plates. The linear stability framework allows one to obtain a map of the ominant moes in the (M,U,)-space, an we observe that, while it is possible to observe a transition from in-phase to out-of-phase for increasing an a particular choice of (M,U ), in some situations the in-phase or the out-of-phase moes are ominant for all, consistent with some recent experiments (Jia et al., 7). In general, the presence of a secon plate was shown to estabilize the straight rest position. The estabilizing effect was observe to be most significant for heavy or short flags (small M ). This preferential estabilization for small M was also observe in Guo an Païoussis (b) for the stability of a single plate in a channel, linking this estabilization to confinement effects. Our analysis also shows that the ominant moe properties are strongly influence by the flui-soli inertia ratio M : the out-of-phase moe is linearly ominant for all values of for heavy plates (small M ), as observe numerically by Tang an Païoussis (9), while the in-phase moe can be ominant for larger values of M. In the case of N = 3 plates, three types of moes (symmetric, in-phase an out-of-phase) were observe, consistently with recent win-tunnel experiments by Schouveiler & Eloy. The case of an infinite number of plates was then iscusse. The stability curve was obtaine as well as the properties of the most unstable moe. The out-of-phase moe (with two consecutive plates having opposite isplacements) was shown to be the most unstable for low M an larger values of. However, moes with any value of the phase ifference ϕ [,π] were shown to exist for some parameter values. The phase ifference of the most unstable moe (or equivalently the number of plates necessary to escribe this moe) was observe to conition the convergence of the stability curve for increasing N. The avantage of the present approach over the one-imensional approach evelope in Jia et al. (7) an Shelley et al. () lies in its ability to take into account the finite length of the plate an the presence of a wake enforcing a trailing ege regularity conition. However, aitional moelling work is neee to inclue a more accurate representation of the vortical wake. From the comparison between the present ouble-wake approach an the vortex sheet representation for one plate, we expect the former to behave relatively well for larger values of M. For shorter or heavier flags, the influence of the wake is expecte to be more important an we anticipate some iscrepancies. Acknowlegements This work was supporte by the Human Frontier Science Program Research Grant RGY 73/.