A simple low order model of the forced Karman wake

International Journal of Mechanical Sciences 52 (2010) 1522–1534

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A simple low order model of the forced Karman wake Njuki W. Mureithi a,n, K. Huynh a, M. Rodriguez b, A. Pham a a b

Department of Mechanical Engineering, Ecole Polytechnique Box 6079, Succ. Centre-ville Montreal, Quebec, Canada H3C 3A7 LadHyX, Ecole Polytechnique, 91128 Palaiseau, France

a r t i c l e in f o

a b s t r a c t

Article history: Received 1 October 2009 Received in revised form 17 May 2010 Accepted 16 June 2010 Available online 10 August 2010

The present work develops a very simple mathematical model for the 2D von Karman vortex shedding wake. The case where this wake is periodically excited at the vortex shedding frequency is considered. The goal is to arrive at an approximate model that is simple enough to allow a full analysis of the underlying nonlinear dynamics. Since such a simple model cannot be expected (and is not intended) to replicate the Navier–Stokes equations the comparison criteria is the ability of the model to replicate the sequence of bifurcations as the forcing amplitude parameter is varied. Such a model can be useful for flow control applications. Equivariant bifurcation theory is employed to obtain the low order discrete model for the dynamics of the Karman wake when ‘reflection-symmetrically’ forced at the vortex shedding frequency. The discrete dynamical system (amplitude equations) modeling the mode interactions is derived in Poincare´ space. Model parameters are then determined via POD from numerical simulations of the simple stationary cylinder case. A quantitative analysis of the wake dynamics based on the model above is presented. For Re¼ 1000, dynamics of the 2D cylinder wake are shown to be closely linked, via the Bogdanov–Takens bifurcation scenario to physical and mathematical systems having SO(2) symmetry. For Re¼ 200, a torus breakdown following the Afraimovich–Shilnikov scenario is found. The result is a complex, possibly chaotic, wake flow. Experimental results, in the form of measured POD modes, are also presented. The results suggest that the 3D wake transition does not destroy the 2D Karman wake dynamics; the latter apparently remains dominant even at moderately higher Reynolds numbers. The goal of the present work is to show that much can be learnt from a very simple model of the wake dynamics when the role of symmetry is carefully considered. This could also be viewed the other way around; that the dynamical bifurcation behavior of the apparently complex forced wake flow can, somewhat surprisingly, be described by a fairly simple model. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Vortex shedding Symmetry-equivariance Low order model Bogdanov-Takens bifurcation

1. Introduction The present work is motivated by the general observation that apparently complex systems often tend to have very simple well defined behavior. The system under consideration is the shear flow which forms the wake behind a bluff body—in this case a cylinder in transverse flow. At a Reynolds number Rec  48, the cylinder wake destabilizes resulting in the well known von Karman periodic wake. What is remarkable of course is the fact that the flow ‘chooses’ such a simple harmonic oscillator behavior following the instability. The existence of such an oscillator can be rigorously proved as first suggested by Landau and Lifschitz [11]. The Landau equation shows that the Karman wake results from a supercritical Hopf bifurcation of the flow at the critical Reynolds

n

Corresponding author. Tel.: +1 514 340 4711; fax: + 1 514 340 4176. E-mail address: [email protected] (N.W. Mureithi).

0020-7403/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2010.06.012

number. More detailed work considering the spatial-temporal problem by Betchov and Criminale [20] showed more specifically that the oscillating Karman wake is a global instability mode manifested following spatio-temporal (absolute) instability over a ‘large’ region of the cylinder near wake. Motivated by the confirmation of simple harmonic behavior, Hartlen and Currie [9] developed a vortex-induced vibration (VIV) model using the harmonic oscillator as a starting point then adding appropriate nonlinear terms. To better understand the wake dynamics, experiments have been carried out to investigate the effect of external periodic forcing on the wake. This problem is particularly important when trying to understand the wake-structure interactions during vortex induced vibrations (VIV). The structural response undergoes important changes as the flow velocity is varied following lock-in (see e.g. [9]). In the forced excitation tests, numerous vortex wake modes have been observed having different spatio-temporal symmetries.

N.W. Mureithi et al. / International Journal of Mechanical Sciences 52 (2010) 1522–1534

These synchronized patterns (modes) are commonly labeled according to the number of vortices shed per cycle of cylinder oscillation, Williamson and Roshko [18]. The S and P modes for instance indicate single vortex and pairs of vortices shed per cycle, respectively. Others include the 2S, 2P, P+S and 2P+ 2S modes. It has been found that the transitions between these wake modes can explain some of the observed phenomena during VIV. For example, as first observed by Williamson and Roshko and then confirmed by Brika and Laneville [2,3], the jump in the response amplitude, which coincides with a half period (p) phase shift between the cylinder motion and the fluid force, is due to the transition, near resonance, from the 2S mode of vortex shedding to the 2P mode. Ongoren and Rockwell [16] have also described a competition between different vortex wake modes. Spatial symmetry was found to be a key feature in the wake flow. More recent work on forced wake dynamics may be found in Baranyi [1] and Didier and Borges [6] among others. The apparently important role played by symmetry prompted Mureithi et al. [12,13], Mureithi and Rodriguez [14] and Mureithi [15] to delve deeper into the implications of the observed wake symmetry in helping understand the underlying wake dynamics. The role and implications of symmetry in fluid mechanics is discussed in the excellent review by Crawford and Knoblock [5]. In Mureithi et al.’s work, a pair of simple discrete amplitude equations governing the nonlinear interaction of the Karman and reflection-symmetry modes was derived using symmetry equivariant bifurcation theory. A qualitative analysis of these equations showed that a number of standard bifurcations of the Karman mode could be expected as the amplitude of the reflectionsymmetric mode was varied. Possible changes in the wake symmetry induced by increased forcing such as period-doubling or symmetry breaking in the cylinder’s wake in the case of streamwise harmonic forcing were predicted. The present work builds on this previous effort to employ symmetry-equivariant theory to the study of the forced Karman wake. The importance of symmetry equivariance is first demonstrated by a simple but classical problem in mechanics—that of the buckling beam. This is followed by a detailed outline of the derivation of the low order model. To clearly demonstrate the physical implications of the resulting low order amplitude equations, general standing wave (SW) and travelling wave (TW) solutions of the resulting differential equations are first presented. In recent work, Rodriguez and Mureithi [17] have performed CFD computations of the wake flow behind a cylinder undergoing periodic excitation in the flow direction. The role of spatialtemporal symmetry in the wake flow was elucidated by a POD analysis of the wake flow velocity field. The modes are compared here with experimentally determined modes to highlight, in particular, the relative importance of 2D versus 3D effects in the forced Karman wake. Finally, the POD modes provide a new avenue for the analysis of the forced wake flow. In the present work, these POD modes are used to determine the unknown coupling coefficients in the amplitude equations. A more quantitative analysis of this low dimensional wake model therefore becomes possible.

2. Symmetry and low order amplitude equations Alternate vortex shedding behind a circular cylinder in transverse flow is an unsteady flow phenomenon resulting from a dynamic instability of the flow around the bluff body at a Reynolds number of approximately 48. From a dynamics point of view, the dynamic instability is the result of a Hopf bifurcation, associated with a pair of purely imaginary eigenvalues. This is but

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one of the many possible bifurcations possible in dynamical systems. A second bifurcation, associated with a zero eigenvalue, is the pitchfork bifurcation. The bifurcation is physically manifested as buckling in axially loaded beams for instance. The pitchfork bifurcation occurs in systems having reflection symmetry. At the bifurcation point, an initially stable (and symmetrical) equilibrium position or state of the system becomes statically unstable. At the same time, two new stable equilibria are born. For a beam restricted to a 2D plane, two buckled states appear on either side of the unstable undeformed state, i.e. the new states of the beam after buckling are related by reflection symmetry. This somewhat obvious result is a manifestation of a fundamental law. This is the law of conservation of symmetry in dynamical systems. Much like mass and momentum in mechanics, symmetry is conserved when system dynamics change. Indeed the laws of mass and momentum conservation can be reduced to symmetry conservation. As is shown below, the basic form of the equations governing dynamical system behavior can be derived by simply invoking symmetry conservation. This possibility has very interesting and useful consequences. The first is that the simplest essential form (known as the normal form, Guckenheimer and Holmes [8]) of governing equations under specified symmetry conditions can be obtained. The second is that independently of the physical system, the resulting dynamics (solutions) from these equations will have the same generic behavior. Hence, for instance, systems having reflection symmetry (which we represent as Z2(k) below) can potentially ‘buckle’ or undergo a ‘divergence instability’—and indeed this is the most likely instability. In the next section the normal form equation associated with reflection symmetry is derived to demonstrate the proposed method. 2.1. Symmetry equivariance: a simple example The present work exploits the naturally occurring symmetry in physical systems to arrive at a simplified mathematical model for the Karman wake flow dynamics. In this section, we introduce the idea of relating physical symmetry to governing dynamical equations. For clarity, we demonstrate the approach with the simple problem of a beam subjected to axial loading. Despite its simplicity, this problem contains all the essential elements of the method applied later to the wake flow problem. The axially loaded beam, Fig. 1, when limited to a plane, has a simple reflection symmetry about a line through the center of the beam and parallel to the beam axis. The symmetries of a given physical system form a mathematical group. To complete the group one must include the identity element, I which is the ’do nothing’ operation. The reflection symmetry group, usually denoted by Z2(k) therefore has two elements, Z2(k)¼{I, k}. The physical presence of the reflection symmetry dictates that the governing equations must also have this symmetry. Reflection about the beam axis corresponds to replacing w by –w in the beam equation of motion. This operation does not change the resulting equation as is demonstrated below. The governing equation is said to be equivariant with respect to the symmetry group Z2(k). It is this symmetry equivariance that makes it possible to determine the general form that an equation or function must

Fig. 1. Beam under axial loading.

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have once the symmetry is known. Symmetry restricts the form that a given function can take. Consider the governing equation (1) for a beam subjected to an axial load P (see for instance, [4]). Here w(x, t) is the dimensionless beam transverse displacement and t and x dimensionless time and length, respectively. This equation has Z2(k) symmetry since reflection k(w)-  w leaves the equation unchanged. To determine the beam equilibrium position as a function of the axial load P the steady version of the equation shown in Eq. (2) is sufficient:   @2 w @2 w @4 w @2 w @w 2 EI 3EA ð1Þ  þ b  s ¼ 0, b ¼ 2 , s ¼ 2P @t 2 @x2 @x4 @x2 @x PL 

  @2 w @4 w @2 w @w 2 þ b  s ¼0 @x2 @x4 @x2 @x

ð2Þ

In the simplest case of a one mode approximation, a Galerkin projection w(x,t)¼ f(x)q(t) yields the following simple steady state equation:

bðPÞqsðPÞq3 ¼ 0

ð3Þ

where

bðPÞ ¼

Z

L





bðPÞf0000 f00 f dx, sðPÞ ¼

0

Z

L

sðPÞf00 fu2 f dx

0

Eq. (3) is an equation of the form F(q, P)¼0. The equilibrium positions are the zeros or roots of the function F(q, P). This brings us to the crucial point of the present work. This is the observation that one can arrive at the essential mathematical form of Eq. (3) or the function F(q, P) based solely on knowledge of the underlying symmetry of the problem. To demonstrate this we start by defining the mathematical representation of symmetry for a given function g(x). The invertible matrix gARn  n is called a symmetry of the (vector) function g(x)ARn if g[g(x)]¼ g(g[x]) (see [7]). In this case the function g(x)is said to be g-equivariant. For the buckling problem above, the function F(q, P) is Z2(k)-equivariant. Since F(q, P) is a scalar function the symmetry elements will also be scalars instead of matrices, hence k ¼  1 and the identity element I¼ 1. For the element k ¼  1 this means that

k½Fðq,PÞ ¼ Fðk½q,PÞ

ð4aÞ

Or, using the scalar representation above, 1  Fðq,PÞ ¼ Fð1  q,PÞ

ð4bÞ

while for the identity element I¼1, the following result is obvious: 1  Fðq,PÞ ¼ Fð1  q,PÞ

ð5Þ

Given conditions (4) and (5) (the latter being trivial) the question is what is the most general possible polynomial form of the function F(q, P)? In this case it is clear, by inspection, that to satisfy these symmetry conditions, F(q, P) must be an odd function of q. Thus it must be of the form: Fðq,PÞ ¼ a1 qþ a2 q3 þ   

ð6Þ

where the coefficients ak are functions of P. Considered up to cubic terms (as also done when deriving Eq. (3)), we note that Eq. (6) has exactly the same form as Eq. (3). The key point here, is that Eq. (6) was ‘derived’ based only on the symmetry of the problem. The simple cubic equation is the so-called normal form of the equation governing the stability behavior of the equilibrium state of all reflection symmetric systems. This highlights another key point of the approach taken here—the generalization ability of symmetry based methods. Hence, the equilibrium state of all reflection-symmetric systems can be reduced to this simple cubic

normal form. And, furthermore, as the parameter P is varied, a pitchfork bifurcation of the equilibrium state will (normally) occur. Eq. (3) has three solutions. The zero solution corresponds to the undeformed state of the beam. Note that this is the only solution for PoPc, indicating that the beam has a single equilibrium state below the critical buckling load Pc—as expected. For P4Pc, this state becomes unstable and two new stable equilibria are born. Clearly, the two new equilibrium states (solutions) no longer have the symmetry Z2(k). The buckling instability is therefore said to ‘break’ the reflection symmetry of the system. However, much like energy and momentum, symmetry must always be conserved. Symmetry conservation is, however, more subtle. When symmetry breaking occurs, one way that symmetry is conserved is by the appearance of multiple solutions, such that the symmetries of all the solutions ‘add up’ to the original symmetry of the system. The symmetry of each particular solution will be a mathematical ‘subgroup’ of the original symmetry. Hence, knowing the subgroups of a given symmetry we can immediately determine the number of possible solutions and their symmetries. Because of symmetry conservation, these solutions are not fully independent but are rather tied together by the system symmetry group, G. It can be shown that if g C G is a symmetry and q a solution, then g(q)is also a solution. One can verify this trivially for the buckled beam—the two possible buckled states are mirror images or related by the symmetry k. The foregoing is a simple demonstration of the application of symmetry equivariance theory to derive the fundamental form of the equation governing the stability behavior of a dynamical system. In the beam buckling example, the spatial reflection symmetry, Z2(k), the simplest possible symmetry, is the underlying symmetry. The mathematical formalism underlying the approach is the Hilbert-Weyl theorem, Golubitsky et al. [7]. Interpreted informally, the theorem states that for any given symmetry (group) G, there exists a finite set of G-equivariant polynomials which form a basis for all G-equivariant functions. Note that the G-equivariant polynomials are essentially the low order models that we seek, expressed in polynomial form. Symmetry equivariant theory is presented in great detail by Golubitsky et al. [7]. 2.2. Flow symmetry We turn now to the problem of wake flow modeling. We start by defining the wake symmetry properties which are important for the study of the wake flow behind a cylinder in cross flow. The near wake flow is idealized as follows. Prior to the initial Hopf bifurcation (which leads to the onset of Karman vortex shedding near Rec  48), the flow has reflection symmetry, about the x-axis (parallel to the upstream flow vector). At some distance downstream of the cylinder the wake is also (locally) x-translation invariant, Fig. 2. Let the local flow velocities in the x- and y-directions be u(x, y, t)and v(x, y, t), respectively. The 2D flow symmetries above mean the following relations hold: uðx,y,tÞ ¼ uðx þ l,y,tÞ uðx,y,tÞ ¼ uðx,y,tÞ vðx,y,tÞ ¼ 0

ð7Þ

Fig. 2. Idealized wake flow with symmetry G ¼ Z2(k)  SO(2) ¼ O(2) in the dashed rectangle.

N.W. Mureithi et al. / International Journal of Mechanical Sciences 52 (2010) 1522–1534

The reflection symmetry is here also represented as Z2(k), while the translation symmetry is represented by SO(2). The overall symmetry is therefore G ¼Z2(k)  SO(2). G is therefore simply the orthogonal group, O(2), the symmetry of the plane. Instability and transition from a flow state with symmetry G to a state with ‘less’ symmetry involves symmetry breaking; the resulting flow symmetry must, however, be a subgroup of G. When symmetry is broken multiple solutions of lower order symmetries appear such that the summation of all the lower order symmetries adds up to the original base symmetry; thus ‘overall’ symmetry is conserved via distribution over several solutions. Unlike the case of the buckling beam, where Z2(k) is a two element discrete symmetry, here the symmetry O(2) is a continuous symmetry group. This means that there is potentially an infinite number of solutions when symmetry is broken. Certain symmetries are, however, privileged and are most likely to appear. These are the so-called maximal isotropy symmetry subgroups. Maximal isotropy subgroups have the unique property that they are the symmetries most likely to appear following instability from the symmetry G. For the O(2) symmetric wake flow these subgroups are Z2(k, p) and Dm. The Z2(k, p) is a two element spatio-temporal symmetry. The first element is of course the identity element. The second element is a combined reflection plus half period translation in time. The symmetries Z2(k, p) and Dm (with m¼2) are the symmetries of the well known sinuous and cosinuous ‘unstable’ modes found in the wake of a bluff body. The sinuous mode is better known as the Karman wake shedding mode. We label this mode K while the reflection symmetric mode is labeled S. The two modes are schematically depicted in Fig. 3. The mode S velocity field satisfies the relations: uðx,y,tÞ ¼ uðx,y,tÞ ¼ uðx þ lS ,y,tÞ ¼ uðx,y,t þ tS Þ vðx,y,tÞ ¼ vðx,y,tÞ ¼ vðx þ lS ,y,tÞ ¼ vðx,y,t þ tS Þ

ð8Þ

(Eq. (10)) are expressed relative to the wavelength of mode K; hence

p represents a spatial translation by half the wavelength of this model i.e. lK/2. ‘m’ is then the wavelength ratio lK/lS.

3. Symmetry equivariance and low order model equations In the present work we consider the effect of external perturbations on the Karman shedding mode. The external perturbations are represented by a reflection-symmetric S mode. To model the resulting dynamics, Karman shedding is represented by the mode K. We can investigate the effect of interaction of the Karman mode with reflection symmetric modes S of different wavelengths by varying the wavelength ratio m. The results presented here, however, are for the case m¼1. Our primary goal then is to study the resulting interaction between modes S and K of equal wavelength. As a first approximation we describe the interaction in terms of the respective modal amplitudes. Assuming that all other modes are stable, the x-direction velocity perturbations, for instance, may be expressed as uðx,y,tÞ ¼ SðtÞcS ðyÞeiðlS x þ os tÞ þ KðtÞcK ðyÞeiðlK x þ oK tÞ þ complex conjugate

ð11Þ where S(t) and K(t) are the mode amplitudes for modes S and K, respectively. A similar expression can also be written for the y-direction velocity perturbations. 3.1. Symmetry equivariance We wish to determine next how the different symmetries act on the mode amplitudes S(t) and K(t). We start by noting that u(x, y, t) is reflection symmetric, in other words

k3uðx,y,tÞ ¼ uðx,y,tÞ uðx,y,tÞ ¼ uðx,y,t þ tK =2Þ ¼ uðx þ lK =2,y,tÞ ð9Þ

The appropriate wavelength and period are represented by l and t, respectively. For the Karman K mode for instance, tK ¼ 2p=oK where oK is the vortex shedding frequency. Mode S therefore has purely (dihedral group) spatial symmetry GS. The order of this group is, however, defined based on wavelength ratio lS/lK ¼ m. As seen in Eq. (9) mode K possesses a mixed spatio-temporal symmetry GK as described above. The symmetries may be compactly expressed as

GK ¼ Z2 ðk, pÞ  S1 

GS ¼ Dm k,

 2p  S1 m

ð10Þ

Note that the circular S1 symmetry appears as a consequence of the periodicity associated with the Hopf bifurcation. The symmetries

Fig. 3. The S and K wake flow modes.

ð12Þ

The reflection symmetry element k acts on the velocity profiles cS(y) and cK(y) as follows:

For the K mode, on the other hand vðx,y,tÞ ¼ vðx,y,t þ tK =2Þ ¼ vðx þ lK =2,y,tÞ

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k3cS ðyÞ ¼ cS ðyÞ k3cK ðyÞ ¼ cK ðyÞ

ð13Þ

For Eq. (13) to hold and in view of Eq. (11), we deduce that the action of Z2(k) on the amplitudes S(t) and K(t) must be

k3ðS,KÞ ¼ ðS,KÞ

ð14Þ

One can similarly show that the symmetry SO(2) acts as

y3ðS,KÞ ¼ ðeimy S,eiy KÞ

ð15Þ

where y acts via the translation x-x+ (lKy/2p) and m¼ lK/lS Z1 is assumed to be an integer. 3.2. Normal form equations and low order model We are interested in mode interaction where mode K and mode S wavelengths are related by an integer ratio m. In the present work we consider the case m ¼1. This means that the mode S considered here is an artificial ‘mode’ consisting of symmetrical flow perturbations at the vortex shedding frequency. Our aim is to derive the general mathematical form of the amplitude equations governing the nonlinear interactions between these two ‘modes’. The symmetry equivariance properties will be used. For convenience we introduce a discrete Poincare map. The mapping, sampled at the mode K frequency, allows us to implicitly ‘discard’ the circular symmetry S1 associated with time periodicity which appears in Eq. (10). The periodicity must, however, be considered when discussing the results; hence, for instance, a periodic

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solution in the discrete Poincare space actually corresponds to a double frequency solution on a torus due to this periodicity. The evolution of the fluid state may be represented by a mapping of the form: un þ 1 ðx,y,tn þ 1 Þ ¼ F½un ðx,y,tn Þ

ð16Þ

Eq. (16) states that the velocity at step n+1 is a function of the velocity at the preceding step n, where the two states are one period apart (corresponding to the Poincare map sampling period). This equation is essentially a very compact representation of the Navier– Stokes equations. Note, however, that here we do not derive the explicit form of the function F[ ] from the Navier–Stokes equations. Instead, the symmetry equivariance approach introduced above will be used to arrive at a compact polynomial approximation. By introducing the two-mode approximation, Eq. (11), we may reduce Eq. (16) to the pair of discrete amplitude equations:

ð17Þ

The functional forms of the mappings f1 and f2 are next determined based on knowledge of the symmetry actions (14) and (15) on the amplitudes Sn and Kn. Recall that the function F(S, K) ¼[f1(S, K), f2(S, K)]T is G-equivariant (or G-symmetric) if

g3FðS,KÞ ¼ Fðg3S, g3KÞ 8 g A G

ð18Þ

For Z2 ðkÞ using Eq. (14) we obtain

ð19Þ

Eq. (19) confirms the expected result that f1(S, K) is an even function of K, while f2(S, K) is odd. The action of an element eimyof the translation symmetry is eimy f1 ðS,KÞ ¼ f1 ðeimy S,eiy KÞ eiy f2 ðS,KÞ ¼ f2 ðeimy S,eiy KÞ

For the case where the S mode is associated periodic excitations in the flow direction at the vortex shedding frequency, thus lK/lS ¼1 the resulting (inflow) u-velocity and (transverse) v-velocity flow field fluctuations may be expressed by the following one mode approximation:

~ ðyÞeiðlK x þ oK tÞ ~ c vs ðx,y,tÞ ¼ SðtÞ S

ð22Þ

At the linear scale, the same flow field would be generated for Re448 but now this field would interact with the (already present) Karman shedding mode K. Based on the lemma above, for this case lK/lS ¼1 (which means oS ¼ oK), hence m¼ n¼1, the amplitude equations are: dK 2 2 ¼ ða0 þ g11 9S9 þ a2 9K9 ÞK þ d01 S2 K dt

f1 ðS,KÞ ¼ f1 ðS,KÞ f2 ðS,KÞ ¼ f2 ðS,KÞ

4. Mode interaction dynamics

us ðx,y,tÞ ¼ SðtÞcS ðyÞeiðlK x þ oK tÞ

Sn þ 1 ¼ f1 ðSn ,Kn Þ Kn þ 1 ¼ f2 ðKn ,Sn Þ

only perturbations with the symmetries considered here. We remark, however, that the derivation here assumes reasonably small amplitude perturbations. Recall also that the symmetries considered here are only approximate over a limited region of the wake flow hence, in the far wake or very close to the cylinder, other effects may become dominant.

ð20Þ

dS 2 2 ¼ ðb0 þ b2 9S9 þ g21 9K9 ÞS þ m01 SK 2 dt

ð23Þ

where an over-bar indicates complex conjugation. The equations are presented here in differential form in accordance with the traditional representation of amplitude equations in fluid dynamics. Eq. (23) may be conveniently expressed in polar form by introducing the relations KðtÞ ¼ rðtÞeifðtÞ and SðtÞ ¼ qðtÞeiZðtÞ . The resulting equations are:

3.3. Normal form equations

dr ¼ ða0 þ a2 r 2 þ g11 q2 Þr þ d01 rq2 cos 2y dt

Eqs. (19) and (20) are the constraints on the generally complex functions f1 and f2 that must be met in order to satisfy the symmetry conditions. Based on these constraints, the lemma below gives the general mathematical form of the complex function (mappings) F(S, K)¼[f1(S, K), f2(S, K)]T. Note that in the lemma, the more general wavelength ratio lK/lS ¼m/n is considered.

dq ¼ ðb0 þ b2 q2 þ g21 r 2 Þq þ m01 r 2 q cos 2y dt dy ¼ qrðm01 r þ d01 qÞsin 2y dt

ð24aÞ

where the relative phase y ¼ Z  f. Note that the individual phase derivatives are given by

Lemma. (Mureithi et al. [13]). Every G ¼Z2(k)  SO(2) equivariant map F : C 2 -C 2 has the form 2 3 n1 m pðr1 ,r2 ,r3 ÞS þqðr1 ,r2 ,r3 ÞS K 5 ð21aÞ FðS,KÞ ¼ 4 m1 rðr1 ,r2 ,r3 ÞK þ sðr1 ,r2 ,r3 ÞSn ,K

df ¼ d01 rq2 sin 2y dt dZ ¼ m01 qr 2 sin 2y dt

for m¼2k, and 2 3 2n1 2m pðr1 ,r2 ,r4 ÞS þqðr1 ,r2 ,r4 ÞS K 4 5 FðS,KÞ ¼ 2m1 rðr1 ,r2 ,r4 ÞK þ sðr1 ,r2 ,r4 ÞS2n ,K

Due to the phase coupling the pair of complex equations (23) is reduced to three equations instead. Eq. (24) (or 23) are the amplitude equations governing the nonlinear interaction of the two modes. These equations show that the interaction between the modes appears first at only third order terms; the modes do not interact linearly or at second order. This is completely dictated by the underlying symmetries. Despite the complexity of these equations, one can immediately identify two types of solutions. First, standing wave (SW) solutions which correspond to fixed point (constant amplitude) solutions of Eq. (24). These solutions satisfy the condition sin 2y ¼ 0. The fixed point amplitudes are solutions of the

ð21bÞ

where p, q, r, s are polynomial functions of the G-invariants r1 ¼9S92, m r2 ¼9K92 and r3 ¼ Sn K or r4 ¼ r32 and the complex conjugate. The lemma above gives the general polynomial form of the amplitude equations governing the nonlinear interaction of modes K and S when the two modes have wavelength ratio m/n. Note that these do not have to be proper modes of the flow,

ð24bÞ

N.W. Mureithi et al. / International Journal of Mechanical Sciences 52 (2010) 1522–1534

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following algebraic equations:

4.1. Computed wake POD modes

dr ¼ 0 ¼ ða0 þ a2 r 2 þ g11 q2 Þr 7 d01 rq2 dt

In the present work we are considering the simpler problem of forced excitation of mode K by mode S. This means that in the model (23) and (24) the mode S amplitude becomes a known control parameter proportional to the cylinder forcing amplitude. The unknown model parameters are then only those of the K mode equation. To determine these parameters, numerical simulations or experimental measurements of the base flow must be done. Here we choose to use the numerically computed flow around a stationary cylinder. The Reynolds numbers of interest are Re¼200 and 1000. The 2D numerical computations were performed using the commercial code FLUENT. The rectangular domain around a 20 mm diameter circular cylinder, had boundaries 15D upstream, 40D downstream and 20D on the lateral sides. A structured mesh with 260 000 elements (corresponding to 125 000 nodes and 135 000 cells) for Re¼1000 and 120 000 elements for Re ¼200 was used. To solve the Navier–Stokes equations, a second order upwind scheme was used to obtain the face values needed for the integral computations while a second order temporal integration was performed during the simulations. Modeling of the forced oscillating cylinder was done using time dependent inertia forces and boundary conditions to simulate the problem in the nonGalilean cylinder reference frame. The simulation time step was of the order of 1/100 of the average vortex shedding frequency. Mesh convergence tests were performed to ensure accuracy of the results. Further details may be found in Rodriguez [19]. While it is well known that 2D simulations for Re4200 do not capture the 3D cylinder wake flow, we remind the reader that the present work investigates the effect of 2D perturbations on the wake flow. The question of 3D effects is revisited and dealt with based on experimental measurements which are obviously 3D. The computation results showed that, as expected, for Re ¼200 and 1000, the flow is dominated by the Karman mode K followed by a weaker S mode (at twice the shedding frequency). To extract these modes from the flow field data, a proper orthogonal decomposition (POD) of the data was performed. POD may be loosely interpreted as the spatio-temporal equivalent of a modal decomposition in vibrations. The difference here being that the ‘modes’ are functions of space and time. The analysis is based on the Karhunen–Loe ve decomposition. A helpful introduction to POD analysis and its application in fluid dynamics may be found in Holmes et al. [10]. The POD analysis was performed to extract the principal modes in the streamwise velocity profile u(y, t) at locations downstream of the cylinder. In the computations, the flow field is projected onto an orthonormal set of functions Ck(y), (topos), each having time evolution ak(t) (chronos). The velocity profile may therefore be expressed as

dq ¼ 0 ¼ ðb0 þ b2 q2 þ g21 r 2 Þq 7 m01 r 2 q dt dy ¼0 ð25Þ dt Note that the phase-locked condition sin 2y ¼ 0 leads to the phase angle relation:

c ¼ f þ p2n, n ¼ 0,1,2,3,. . .

ð26Þ

Solution of Eq. (25) yields the standing wave (SW) amplitudes rSW and qSW and phases ZSW ¼ fSW if we arbitrarily take n ¼0. For this solution, the corresponding u-velocity standing wave flow field will be of the form: uSW ðx,y,tÞ ¼ rSW eifSW cK ðyÞeiðlK x þ oK tÞ þ qSW eiZSW cS ðyÞeiðlS x þ oS tÞ   ¼ 2 rSW cK ðyÞ þ qSW cS ðyÞ cosðlK x þ oK t þ fSW Þ

ð27Þ

where we take into consideration the fact that lS ¼ lK and oS ¼ oK. The second type of solutions are travelling (TW) wave solutions. TW solutions correspond to the case dy=dt ¼ 0 but sin 2y a 0. This means that dy ¼ 0 ¼ m01 r þ d01 q dt sin 2y ¼ w ¼ const

ð28Þ

From Eq. (28) we note that 2y ¼ p 7 s and s ¼ sin1 w. Using this result in Eq. (24b) we conclude that dZ ¼ 8 m01 qr2 sin s ¼ 8 m01 qr2 w dt

ð29Þ

Based on Eq. (29) and since y ¼ yTW ¼ const., we must have

ZTW ðtÞ ¼ 8 oTW t p

s

2

2

fTW ðtÞ ¼ 8 oTW t 8

ð30Þ

where the travelling wave frequency oTW ¼ 8 m01 qr2 w. For the travelling wave solution the resulting u-velocity perturbations will have the form uðx,y,tÞ ¼ rTW eifTW cK ðyÞeiðlK x þ oK tÞ þ qTW eiZTW cS ðyÞeiðlS x þ oS tÞ  ¼ 2 rTW cK ðyÞcosðlK x þ ðoK 8 oTW ÞtÞ þ qTW cS ðyÞcosððlK x p si ð31Þ þ ðoK 8 oTW ÞtÞ 8 Þ 2 2 The left and right travelling waves are explicitly seen in Eq. (31). Note that the modal amplitudes are related by Eq. (28). The SW solution (27) and TW solution (31) are the simplest solutions of Eq. (24) since the oscillation amplitude is constant in both cases. More complex solutions with varying amplitude and phase are also possible. The solution actually manifested depends on the numerical values of the constants a, b, g, d and m in the amplitude equations (23) and (24). The solution that is physically manifested is the stable solution. For the basic SW and TW solutions, stability is easily determined by evaluating the Jacobian matrix of Eq. (24) on the solutions (rSW, qSW) and (rTW, qTW), respectively. At this point, and for further detailed analysis, numerical values of the model parameters a, b, g, d and m are needed. These parameters will be determined in the sections that follow.

uðy,tÞ ¼

r X

ak ðtÞCk ðyÞ

ð32Þ

k¼1

The chronos magnitude represents the relative modal amplitude, hence energy contained in the corresponding topos. Fig. 4 shows the topos for the first two modes of the x-velocity u(y, t) on the 10D line downstream of the cylinder. The figure shows contour plots of the u-velocity perturbations corresponding to each mode. The modal decomposition clearly brings out the symmetry in the flow. This spatial-temporal symmetry may be conveniently described by normalizing the basic shedding wavelength to 2p. Normalizing the wavelengths so that the Karman mode wavelength lK  2p, helps characterize the spatiotemporal symmetry of the modes. Thus, respectively, the first three modes found in the fixed cylinder wake have Z2(k, p), D2(k, p) and Z2(k, 2p/3) symmetries. In this notation then, the

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Fig. 4. Computed first and second mode topos 10D downstream of the fixed cylinder for Re ¼1000.

Fig. 5. Tests results for Re¼ 200 showing (a) flow field and (b) contours of the measured (u-) streamwise velocity peturbations.

Karman mode symmetry is Z2(k, p), meaning that this mode is invariant under combined reflection (k) and translation downstream by half a period, see Fig. 4(a). D2(k, p) symmetry means the second mode, Fig. 4(b), is both invariant under reflection and under translation parallel to the flow by half a period. These are the symmetries discussed earlier. Before obtaining the amplitude equation parameters from the POD modes, it is important to verify firstly, whether similar modes are obtained experimentally and secondly whether the underlying 2D flow assumption is valid over at least a reasonable Reynolds number range. The latter point is important since, as the reader might have noticed, the symmetry group considered when deriving the low order model was a 2D planar symmetry. A 2D flow assumption was therefore implicitly invoked.

4.2. Experimental wake POD modes The low order model (23,24) based on a 2D flow assumption raises questions regarding the validity and the applicability of the results to realistic flows, even at low Reynolds numbers. To respond to this concern and to verify the applicability of the model, experimental tests have been conducted in a wind tunnel

Fig. 6. Experimental test results for Re ¼200 showing (a) first POD mode and (b) second POD.

for Re¼200 and 1000, to determine the wake POD modes. In the tests, particle image velocimetry (PIV) was used to measure the velocity field in the cylinder wake. The tests were conducted in a low-speed wind tunnel of test section dimensions 60 cm  60 cm. The temperature controlled recirculating wind tunnel has a maximum empty test section speed of 90 m/s and turbulence intensity below 0.5%. The 76.2 and 19 mm diameter test cylinders spanned the test section eliminating unwanted end effects. Tests results for Re¼200 are shown in Fig. 5. The flow field is shown in Fig. 5(a) while Fig. 5(b) shows contours of the measured (u-) streamwise velocity. At this Reynolds number, the wake flow is still predominantly 2D and the periodic wake structure highly organized. The first two modes from a POD analysis of this experimental flow field are shown in Fig. 6(a) and (b). Together these two modes contain over 90% of the energy as determined based on the corresponding singular values shown in Fig. 7. At Re¼200 the wake is clearly still predominantly 2D.

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Fig.7. Experimental POD mode singular values for Re ¼200.

Fig. 9. Experimental tests for Re ¼1000 showing (a) first POD mode and (b) second POD.

Fig. 8. Experimental tests for Re¼ 1000 showing (a) wake flow and (b) v-velocity contour.

Test results for the higher Reynolds number Re¼1000 are shown in Figs. 8–10, here for the transverse velocity v. At this Reynolds number, 3D effects are clearly apparent in the more complex flow structure, Fig. 8. The first POD, Fig. 9(a) is, however, still clearly dominant and highly structured. The second mode, Fig. 9(b) is also reasonably well defined, even when viewed on a single 2D plane. The two modes contain over 80% of the flow energy as shown by the singular values of Fig. 10. Even at Re¼1000 the first two modes predominate the flow. These modes also closely resemble the corresponding modes for Re¼200. From a dynamics viewpoint, the fact that the 2D structures remain relatively intact at Re¼ 1000 is very important. The amount of energy they carry indicates that the modes can be expected to still dominate the wake dynamics. Hence, even if quantities such as

Fig.10. Experimental POD mode singular values for Re¼ 1000.

lift and drag forces are somewhat modified by 3D effects—these modes still drive the fundamental dynamics. The experimental results above therefore suggest that the 2D based low order modeling approach is not wholly unrealistic at low Re. We remark that the experimental results are for the stationary cylinder, hence unforced wake. Wake forcing correlated along the cylinder span, as considered in this work, can be expected to lead to increased correlation along the cylinder axis. More importantly, note that the model presented here is based on the interaction between the first POD mode and a second ‘mode’ associated with 2D perturbations due to cylinder motion.

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5. CFD based mode interaction dynamics and comparison with model prediction In this section, the dynamics of the periodically excited wake are presented. The dynamical behavior is quantified via wake flow stability and bifurcation behavior. First, the results obtained via CFD based numerical solution of the Navier–Stokes equations are presented. This is followed by an application of the low order model. The goal is to demonstrate the capability of the model to predict the dynamics and bifurcations in the wake flow. 5.1. Cylinder-wake response to external forcing for Re¼1000 The effect of cylinder streamwise forcing on the wake flow was studied in terms of vortex wake modes as a function of the cylinder oscillation amplitude. The inline forcing was at the average of the Karman shedding frequency. Five response ranges were exhibited as the perturbation parameter was increased. For small cylinder amplitudes, up to A/D ¼0.075 (D being the cylinder diameter), the wake has a quasi-periodic response. At A/D ¼0.175, the wake destabilizes via reflection-symmetry breaking into a P +S configuration—pair of vortices followed by single vortex shed per forcing cycle. The resulting wake structure is shown in Fig. 11 for A/D ¼0.25. The breaking of reflection symmetry here is fundamentally the identical to the symmetry breaking associated with the beam buckling instability. In both cases two new solutions, related by reflection symmetry, Z2(k), are obtained. Changing initial conditions yields a mirror image of the solution of Fig. 11. For A/D¼0.35, Fig. 12, the wake shedding mode has a configuration similar to the Karman wake. However, vortex shedding occurs at half the Karman frequency; the corresponding Strouhal number is therefore near 0.1. From a dynamics view point, this indicates that a period-doubling bifurcation of the wake flow has occurred.

Fig. 13. Forced wake for a cylinder amplitude of A/D ¼ 0.35 and Re¼ 200.

5.2. Cylinder-wake response to external forcing for Re¼200 For Re¼200, similar results are obtained, but with some important differences. Significantly larger cylinder amplitudes are needed to trigger wake flow bifurcations at this lower Reynolds number. For a forcing amplitude of A/D ¼0.35, a complex chaoticlike wake structure is found as shown in Fig. 13. Note the complex variation in the lift coefficient, in Fig. 13(b), for this chaotic state. For A/D ¼0.5, the system returns to a quasi-periodic state, Fig. 14. This state is, however, quite different from the (quasi-) periodic dynamics on a torus for A/D¼0. In Fig. 14 the system has

Fig. 14. Forced wake for a cylinder amplitude of A/D ¼ 0.5 and Re¼ 200.

Fig. 11. Vorticity field of the flow past the cylinder under forced oscillation for Vr ¼5, A/D ¼0.25 and Re¼ 1000.

Fig. 12. Vorticity field of the flow past the cylinder under forced oscillation for Vr ¼5, A/D ¼ 0.35 and Re¼ 1000.

Fig. 15. Forced wake for a cylinder amplitude of A/D ¼ 0.75 and Re¼ 200.

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undergone a torus-doubling bifurcation. This torus-doubled state has important physical implications. The shedding Strouhal number is now 0.1 instead of the well known value of 0.2 for the stationary cylinder. This was the same results observed for Re¼1000 above. However, the period-doubling is here associated with a torus as opposed to a fixed point for the higher Reynolds number. For a large amplitude of A/D ¼0.75, the torus-doubled state is replaced by a chaotic state as seen in the wake structure and perturbation velocity spectrum of Fig. 15.

5.3. Predicted mode interaction dynamics The low order model (23,24) is now applied to the forced wake problem in this section. The goal is to quantify, from a dynamics point of view, the bifurcation sequence observed above. We suppose that the lowest spatio-temporal mode K dominates the dynamics. Inflow cylinder oscillations are assumed to excite a reflection-symmetric mode (which we call S) having the same wavelength as the K mode. The spatial symmetries of the modes considered are therefore GK ¼Z2(k, p) and GS ¼D2(k, 2p). Since POD gives normalized topos, the amplitude evolution of the modes is contained in the chronos. The map resulting when mode S is considered ‘‘constant’’ has Z2(k, p) symmetry. Since the mode S amplitude is externally controlled, only the parameters of the mode K equation need to be determined. The Poincare´ reduction of the computed spatio-temporal mode data for A/D ¼0 are used. The data are first ‘‘complexified’’ via a Hilbert transform. The average wake period is used for discretization. K and S amplitudes are calculated for 45 wake shedding periods, thus amplitude equations constants can be determined using a least square method to solve the over-determined system of equations. As an example, for Re¼200, the following parameters are obtained:

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and POD amplitude has not been established hence only the bifurcation types can be compared quantitatively.

5.3.1. Low order model dynamics: Re¼1000 Qualitatively, the mapping (23,24) reproduces the observed bifurcations reasonably well. In the numerical computation, we start off in the fixed cylinder case with a closed curve, Fig. 16(a), showing the quasi-periodic nature of the wake dynamics relative to the fixed average shedding frequency. This result is reproduced by the model on the Kr, Ki plane, Fig. 16(b). This initial limit cycle results from the Hopf bifurcation responsible for the onset of vortex Karman shedding. For a forcing amplitude A/D ¼0.05, the limit cycle has undergone a symmetry-breaking bifurcation, Fig. 16(c). For the low order model, this was found to be a pitchfork bifurcation, Fig. 16(d). Increasing the forcing amplitude in the CFD based map, complex, chaotic-like behavior is found for A/D ¼0.075  0.1,

a0 ¼ 0:67240:6042j, a2 ¼ 6:7879 þ6:6528j g11 ¼ 61:30680:6855j, d01 ¼ 0:85931:6349j It is important to clearly define the intended goal of the comparison to be done here. Recall that what has been developed is a very low order (discrete) model of the amplitude equations governing the nonlinear interactions of modes K and S. In the specific case where numerical parameters have been computed, we wish to test the predictive ability of the model for the case where mode K is periodically forced by mode S (the latter assumed to be unaffected by mode K). To verify the model, we wish to compare the sequence of bifurcations predicted by the model (as the mode S amplitude is increased from zero) with the bifurcations obtained from CFD solutions of the Navier–Stokes equations (i.e. the exact model). The bifurcation behavior of a given dynamical system is strongly dependent on the dominant nonlinear terms. A mathematical model which yields the correct sequence of bifurcations is there deemed to be quantitatively very accurate since the stability behavior of the system can be known from an analysis of the model. The model may even be deemed exact—from a topological view point. This means that exact solutions could (in principle) be obtained via a ‘simple’ parameterization of the model solutions. The model is only qualitative because it does not contain the exact parameterization. Note, however, that it is the ‘elimination’ of this parameterization which leads to the simple (general) model in the first place. In the present case, the system dynamics are projected onto the first two modes using only the u-velocity fluctuations. Model comparison is done on the Poincare´ map of the flow. For the numerical data the Poincare map is obtained by sampling the CFD lift coefficient at the average vortex shedding frequency (which is the K mode frequency). Note that at this point in the work, a direct relation between the lift coefficient amplitude

Fig. 16. Low order model prediction relative to CFD simulations showing (a, b) computed and predicted limit-cycles (c, d) computed and predicted pitchfork bifurcation.

Fig. 17. Low order model prediction relative to CFD simulations showing (a, b) computed and predicted chaotic state (c, d) computed and predicted perioddoubling bifurcation.

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Fig. 17(a). A homoclinic bifurcation is predicted to occur next by the mapping 17(b). Homoclinic bifurcations are known to be an important precursor to chaos. We hypothesize that this is actually what happens in the higher dimensional fluid systems resulting in the Poincare´ map of Fig. 17(a). The final state is a highly organized period-2 state as shown in Fig. 17(c). The low order model (24) on the other hand predicts a pair of fixed point states appearing via a pitchfork bifurcation. As shown in the appendix, mapping (24) (considering the K mode equation only) is the square of a second, lower order, mapping which is obtained when the presence of GK ¼Z2(k, p) symmetry is considered. A period-doubling bifurcation of this, more fundamental, lower order map translates into a pitchfork bifurcation in the squared mapping. The bifurcation sequence described above is the classical Takens– Bogdanov bifurcation scenario [21]. The hallmark of the Takens– Bogdanov bifurcation is the presence of double-zero eigenvalues for the reduced linearized system. The present system may be viewed as an unfolding of the double-zero eigenvalue system.

5.3.2. Low order model dynamics: Re¼200 Figs. 18–21 show a comparison of the wake dynamics obtained by CFD with the model predictions for Re ¼200. Similarly to the CFD results, low amplitude forcing leaves the Poincare´ map in a quasi-periodic state as seen in Fig. 18(a and b) for CFD and low-order model cases, respectively. This quasi-periodic state undergoes a saddle-node (S-N) bifurcation as shown in Fig. 19(a) from the CFD computations. This bifurcation is correctly predicted by the low order model as seen in Fig. 19(b). The model, being low order gives fewer saddles (S) and nodes (N) in comparison to the full Navier–Stokes equations.

Fig. 18. Limit cycle at zero forcing showing (a) CFD simulation showing lift Poincare map and (b) low order model prediction in terms of K mode amplitude.

Fig. 19. Saddle-node bifurcation from (a) CFD simulation showing lift Poincare map and (b) prediction by low order model in terms of K mode amplitude.

Fig. 20. Period-doubling bifurcation from (a) CFD simulation showing lift Poincare map and (b)prediction by low order model in terms of K mode amplitude.

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Afraimovich–Shilnikov and Takens–Bogdanov – leading to a deeper understanding of the forced wake dynamics. The present model is not intended as a replacement for the Navier–Stokes equations—far from it! Rather, its simplicity makes accessible the key dynamics governing the Karman wake response to forced excitation. The model also links this complex system to other physical systems having the same symmetry. Finally, perhaps the most promising potential application for the present model is in the area of flow control. The model allows one to exploit the known dynamics between the Karman mode and a control mode. One could exploit this relationship to trigger bifurcations of the Karman mode thus curtailing its destructive effect for instance.

Acknowledgement This work was performed with the financial support of the National Science and Engineering Research Council of Canada (NSERC).

Appendix The solution of the equation: Kn þ 1 ¼ FðKn , mÞ,

m ¼ mðSÞ

ðA1Þ

yields the (flow) mapping Fig. 21. Chaotic dynamics from (a) CFD simulation showing lift Poincare map and (b) prediction by low order model in terms of K mode amplitude.

The saddle-node state is next replaced by a period-doubled torus as seen in Fig. 20. Fig. 20(a and b) appear different due to the different Poincare planes selected in the CFD computations and the low order model. In Fig. 20(a) the Poincare plane cuts the torus normal to its larger diameter (hence two small loops). In the model, the Poincare plane is normal to the smaller diameter hence the two large concentric loops. Increasing further the forcing amplitude leads to chaos for the CFD computations, Fig. 21(a). This bifurcation to chaos (via period-doubling) is correctly predicted by the low order model as seen in Fig. 21(b). Note that the chaotic attractors’ structure is different since different physical quantities (lift coefficient versus flow velocity) and planes are used in the phase space representation. The transition to chaos here follows the scenarios proposed by the Afraimovich–Shilnikov theorem, which includes chaos through period-doubling or via saddle-node bifurcations. This once again seems to support the potential predictive capability of the proposed low order model.

6. Conclusion Symmetry group equivariant bifurcation theory has been employed to derive the lowest order dynamical system modeling cylinder wake mode interactions in Poincare´ space. This surprisingly simple discrete map has been shown to yield the main bifurcations observed in CFD simulations at two values of Reynolds number. We remark that experimental tests by the authors [13] have found similar dynamics, including perioddoubling bifurcations, at up to Re ¼5000. The low order model makes it possible to focus on the key dynamics underlying the observed wake behavior. Perhaps most importantly, the present model suggests a link between the observed wake dynamics and well known bifurcation scenarios –

Cnn0 : C-C, Kn0 ðt0 Þ-Kn ðtn Þ,K A C

ðA2Þ

Using the normalized shedding period 2p, we define the Poincare´ map from the flow Cnn0 ,

Cpc ¼ C2kp

ðA3Þ

having arbitrarily set n0 ¼0. The mapping Cnn0 is invariant under the symmetry operation GK ¼Z2(k,p). This means that

Cpc ¼ G½Cpc  ¼ k3Cppc

ðA4Þ

Eq. (A4) has the consequence that the Poincare´ mapping Cpc is the square of the lower order mapping: ~ pc ¼ k3Cpc ¼ Cp C pc

ðA5Þ

thus ~ pc Þ Cpc ¼ ðC

2

ðA6Þ

The Poincare´ maps presented in Figs. 16-21 correspond to the mapping Cpc based on period 2p. Note also that the flow symmetry is restricted to GK ¼Z2(k, p) since the excitation mode S is reduced to a single amplitude parameter. Any 2kp-periodic orbit will correspond to a fixed point of Cpc. A symmetry-breaking bifurcation also yields a fixed point. However, the latter corresponds to a period-doubling for ~ pc . the lower order mapping C References [1] Baranyi L. Numerical simulation of flow around an orbiting cylinder at different ellipticity values. Journal of Fluids and Structures 2008;24:883–906. [2] Brika D, Laneville A. An experimental study of the aeolian vibrations of a flexible circular cylinder at different incidences. Journal of Fluids and Structures 1995;9:371–91. [3] Brika D, Laneville A. Vortex-induced vibrations of a long flexible circular cylinder. Journal of Fluid Mechanics 1993;250:481–508. [4] Chen L-Q, Yang X-D. Nonlinear free transverse vibrations of an axially moving beam: comparison of two models. Journal of Sound and Vibrations 2007;299:348–54. [5] Crawford JD, Knoblock E. Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annual Review of Fluid Mechanics 1991;23:341–87. [6] Didier E, Borges ARJ. Numerical predictions of low Reynolds number flow over an oscillating circular cylinder. Journal of Computational and Applied Mechanics 2008;8:39–55.

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