on three-dimensional instabilities of two-dimensional ﬂows

Under consideration for publication in J. Fluid Mech. 1

On three-dimensional instabilities oftwo-dimensional flows with a Z2 spatio-temporal

symmetry

By H. M. B L A C K B U R N,1 F. M A R Q U E S2 AND J. M. L O P E Z3

1CSIRO Manufacturing and Infrastructure Technology, P. O. Box 56, Highett, Vic. 3190, Australia2Departament de Fısica Aplicada, Universitat Politecnica de Catalunya, 08034, Barcelona, Spain3Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA

(Received 17 May 2003)

A number of two-dimensional time-periodic flows, for example the Karman street wake of asymmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a com-bination of evolution by half a period in time and a spatial reflection leaves the solution in-variant. Floquet analyses for the stability of these flows to three-dimensional perturbations havein the past been based on the Poincare map, without attempting to exploit the spatio-temporalsymmetry. Here, Floquet analysis based on the half-period-flip map provides a comprehensiveinterpretation of the symmetry breaking bifurcations.

1. IntroductionA fascinating feature of many time-periodic flows is that they exhibit some kind of spatio-tem-

poral symmetry. A classic example is the two-dimensional wake of a circular cylinder. Snapshotsof dye or hydrogen bubble visualisations of wake structures taken half a period apart in timeshow a reflection symmetry about the wake centreline, y = 0 (x is the streamwise direction, andthe two-dimensional flow is invariant in z, the spanwise direction), as illustrated in figure 1 (a).The corresponding velocity vectors, half a period apart, are mirror images across the centreline ofthe wake. This spatio-temporal symmetry is present in other flows, e.g. the flow in a rectangularcavity driven by the periodic oscillations of the wall at x = 0 (the left vertical wall in figure 1 b),with the other cavity walls at x = h and y = ±h stationary. This flow has been investigatedexperimentally in Vogel, Hirsa & Lopez (2003) and using numerical Floquet stability analysis inBlackburn & Lopez (2003b). Figure 1 (b) illustrates the spatio-temporal symmetry of a periodictwo-dimensional flow in this geometry. When the x = 0 wall moves upwards, a vortex (a ‘roller’if we include a spanwise extent) forms close to the top of the cavity; half a period later, a rollerappears near the bottom of the cavity. For both flows, at any point in time, a linear transformationKy (reflection in the line y = 0) is equivalent to temporal evolution of a half period.

Formally, the spatio-temporal symmetry, H , of these two-dimensional flows acts on the veloc-ity U(x, t) = (U , V , W )(x, y, z, t) as

HU(x, t) = KyU(x, t + T/2) = (U,−V, W )(x,−y, z, t + T/2), (1.1)

and the base flow is H-symmetric: HU(x, t) = U(x, t). T is the fundamental period of theflow. This period is imposed in the non-autonomous case of the driven cavity, where the forcingprovides two parameters. These are the scaled amplitude Re = Vmax h/ν (Reynolds number)and inverse period St = h2/Tν (Stokes number), where h is the x-extent of the cavity, Vmax

is the maximum velocity of the wall at x = 0, and ν is the fluid’s kinematic viscosity. In the

2 H. M. Blackburn, F. Marques and J. M. Lopez

t = t0

(a)

x

y

//

OO

(b)

x

y

//

OO

t = t0 + T/2

FIGURE 1. Example flows with spatio-temporal symmetries: (a) simulated hydrogen bubble visualizationof a two-dimensional circular cylinder wake for Re = 188.5 at times t0 and t0 + T/2 (t0 is arbitrary, T isthe Strouhal period); (b) velocity vectors of a periodically driven two-dimensional rectangular cavity flow,where T is the period of motion of the left-hand wall.

autonomous case of the wake, the period is dynamically determined; it is a function of Reynoldsnumber, Re = U0D/ν, where U0 is the freestream speed and D is the cylinder diameter. Theaction of H on the vorticity, Ω = (Ωx, Ωy, Ωz), is

HΩ(x, t) = KyΩ(x, t + T/2) = (−Ωx, Ωy,−Ωz)(x,−y, z, t + T/2), (1.2)

and for H-symmetric flows, such as the base flow, HΩ(x, t) = Ω(x, t).Although the base flows are both two-dimensional and two-component, and hence ∂z = 0 and

W = 0, W is included in (1.1) because we will be studying symmetry properties of three-di-mensional instabilities. The spatio-temporal symmetry H is isomorphic to Z2, the cyclic groupof order two; the same operation, applied twice, is equivalent to the identity. By virtue of be-ing two-dimensional and two-component, the flows are also invariant to arbitrary translationsor reflections in z; these two additional symmetries, respectively generators of the SO(2) andZ2 symmetry groups, are, when combined, generators of the O(2) symmetry group (when pe-riodicity in z is assumed). The complete symmetry group of the two-dimensional flows underdiscussion is Z2 × O(2).

2. The half-period-flip mapA standard tool for the analysis of T -periodic flows is the Poincare map. The stability of

perturbations to a limit-cycle flow may be examined by determining their behaviour at successiveperiods nT , n ∈ N, thus exchanging the stability analysis of a limit cycle in a flow for the simplerproblem of the stability analysis of a fixed point of a map. For a ‘flow’ φ(t; x0, t0) (a flow in adynamical systems sense, see e.g. Guckenheimer & Holmes 1986), evolving from a set of initialconditions (x0, t0), the Poincare map of x0 is

x0 7→ P(x0) = φ(t0 + T ; x0, t0). (2.1)

In the present setting, with a spatio-temporal symmetry, one can alternatively use the half-pe-riod-flip map

x0 7→ H(x0) = Kφ(t0 + T/2; x0, t0). (2.2)

These two alternatives are illustrated diagramatically in figure 2. A fundamental point, exploitedin the present work, is that since K2 = I , it follows that P = H2 (Jensen, Golubitsky & True

Three-dimensional instabilities of flows with spatio-temporal symmetry 3

K

Et0

Et0+T/2

φ

φ

H

Px0,P(x0),H(x0)

φ(t0 + T/2; x0, t0)

nn^^^^^^^^

..^^^^^^^

FIGURE 2. The T -periodic Poincare return map P and the half-period-flip map H. E t0 is the Poincaresection of the flow for an arbitrary starting time t 0; Et0+T/2 is another section, displaced in time by thehalf-period T/2. K is a spatial symmetry.

1999; Lamb & Melbourne 1999). The Floquet multipliers µP for P are the squares of those forH, i.e. µP = µ2

H.

3. Floquet analysisThe theory underlying Floquet analysis as applied to the unsteady Navier–Stokes equations

has been previously presented in detail for problems with O(2) spatial symmetry (e.g. Barkley &Henderson 1996; Robichaux, Balachandar & Vanka 1999; Blackburn & Lopez 2003b). Floquetstability analysis studies the evolution of a three-dimensional perturbation u′ to a T -periodic‘base flow’ U : u = U + u′. The linearized equivalent of the incompressible Navier–Stokesequations for an infinitesimal perturbation u′ can be written as

∂tu′ = (∂UN + L)u′, (3.1)

where ∂UN + L represents the linearization (Jacobian) of N + L (the nonlinear operator N

contains contributions from both advection and pressure terms, while the linear operator L cor-responds to viscous diffusion) about the base flow U ; ∂UN is the T -periodic linear operatorobtained from N by replacing Au with u′

·∇U +U·∇u′. The operator ∂UN +L is equivalentto the flow φ in (2.1), (2.2), thus

u′0 7→ P(u′

0) = φ(t0 + T ; u′0, t0), (3.2)

u′0 7→ H(u′

0) = Kφ(t0 + T/2; u′0, t0), (3.3)

where now P and H represent the linearized Poincare and half-period-flip maps of the pertur-bation velocity, respectively. Both U and φ have the same spatio-temporal symmetries, and φ issaid to be equivariant with respect to H-symmetry (1.1) (Golubitsky & Stewart 2002).

Perturbation solutions u′ can be written as a sum of Floquet modes, u(t−t0) = u(t0)eγ(t−t0),

where u(t0) are the T -periodic Floquet eigenfunctions of φ and the constants γ = σ + iω areFloquet exponents. The Floquet multipliers, which define the growth of Floquet modes over theperiod T , are related to the Floquet exponents by µP = eγT . Floquet multipliers for the half-pe-riod-flip map are µH = eγT/2, so that µP = µ2

H. The time-periodic basic state becomes linearly

unstable when one or more Floquet multipliers leaves the unit circle.A convenient means of finding µP and u is to compute the eigenvalues and eigenfunctions of

P : the eigenvalues are the Floquet multipliers and the eigenfunctions are u(t0). Likewise, µH

are the eigenvalues of H, while the eigenfunctions are the same as those for P . Krylov methodsare typically used to compute the discrete eigensystem of P , as described in detail by Tuckerman& Barkley (2000). The same methods are simply adapted to compute the eigensystem of H:instead of integrating perturbations over the complete period T on each iteration [i.e. iterating


P(u′)], they are integrated only over T/2, followed by explicit application of the symmetry K,thus iterating H(u′). A side-benefit is that the computational time required for convergence ofthe eigensystem is typically halved.

We now address the spatial structure of the three-dimensional Floquet instabilities of theZ2 × O(2) flows in question. As a consequence of the two-dimensional geometry and boundaryconditions, and the linearity of (3.1), we can Fourier expand u′ in the z direction and analyzethe stability of each Floquet Fourier mode independently for different spanwise wavelengths λ.Since the base flow is two-dimensional and two-component, two linearly independent expansionfunctions for u′ are

(u′, v′, w′)(x, y, z, t) = (u′ cosβz, v′ cosβz, w′ sinβz)(x, y, t), (3.4)

(u′, v′, w′)(x, y, z, t) = (u′ sinβz, v′ sin βz, w′ cosβz)(x, y, t), (3.5)

where β = 2π/λ is a spanwise wavenumber — these break the spanwise O(2) symmetry forβ 6= 0, and are spatially periodic in z.

When the critical eigenvalue is real, the centre manifold is generically two-dimensional (Mar-ques, Lopez & Blackburn 2003), and is generated by linear combinations of (3.4) and (3.5). Forβ 6= 0, these linear combinations generate a circle of solutions. Physically, different linear com-binations correspond to translations in the spanwise direction. Any of these linear combinationscan be used as the desired eigenfunction in the Floquet analysis.

When the eigenvalues are complex, the centre manifold is generically four-dimensional, andwe have two circles of solutions, each one with linear generators of the form (3.4) and (3.5)(Marques et al. 2003). Both circles are mixed by both time evolution and reflection in the span-wise direction. In this complex multiplier case, both travelling and standing wave solutions arepossible (Blackburn 2002; Blackburn & Lopez 2003a,b).

For all the results discussed here, the underlying spatial discretization employs spectral ele-ments, and the numerical Floquet analysis is based on an Arnoldi method (Barkley & Henderson1996; Tuckerman & Barkley 2000). For further details of the computations carried out for thecircular and square-section cylinder wakes, we refer the reader to Blackburn & Lopez (2003a),and for the oscillatory rectangular driven cavity to Blackburn & Lopez (2003b).

4. Synchronous modesThe synchronous modes are those for which the Floquet multipliers are real and positive,

µP = +1, and which do not introduce new temporal periods: near onset, the instabilities evolvein close synchronicity with the base flow. The long and short wavelength instabilities of the two-dimensional wakes of the circular and square-section cylinders (Williamson 1988; Barkley &Henderson 1996; Williamson 1996; Robichaux et al. 1999), and of the flow in the 2:1 rectangularoscillatory driven cavity (Blackburn & Lopez 2003b) are all of this type. Respectively, these longand short wavelength modes are known as modes A and B, and each type appears to have similarunderlying physical instability mechanisms for all three flows.

The spatio-temporal symmetries of modes A and B for the three flows are as follows (Barkley,Tuckerman & Golubitsky 2000; Blackburn & Lopez 2003b). For the wake flows, mode A isH-invariant, i.e. H acting on the wake mode A leaves it invariant:

Hu′(x, t) = (u′,−v′, w′)(x,−y, z, t + T/2) = (u′, v′, w′)(x, y, z, t) = u

′(x, t). (4.1)

The action of H on the wake mode B does not leave it invariant:

Hu′(x, t) = (u′,−v′, w′)(x,−y, z, t + T/2) = −(u′, v′, w′)(x, y, z, t) = −u

′(x, t). (4.2)

For the cavity flow, the linkage is the opposite: mode A breaks H-symmetry, Hu′(x, t) =

Three-dimensional instabilities of flows with spatio-temporal symmetry 5Mode A, Re = 195 Mode B, Re = 265

t = t0

t = t0 + T/2

FIGURE 3. Vorticity isosurfaces for the synchronous wake modes of the circular cylinder, shown for a 10Dspanwise domain extent, and viewed from the cross-flow direction. Translucent isosurfaces are for spanwisevorticity component, solid surfaces are for streamwise component.

−u′(x, t), while mode B is H-invariant: Hu′(x, t) = u′(x, t). So that for the cavity flow, itis the short wavelength mode B that preserves H-symmetry (1.1).†

Figure 3 shows visualizations of modes A and B for the cylinder wake, obtained through di-rect numerical simulation with restricted spanwise periodic length, at Reynolds numbers slightlyabove onset for the two modes (Barkley & Henderson 1996; Henderson 1997). The non-trans-lucent isosurfaces in the figure are of the (streamwise) x component of vorticity. For a H-symmetric flow, from (1.2), the x-vorticity changes sign with t → t + T/2 and y → −y atany fixed (x, z). The figure, with views in the (cross-flow) y direction, shows this to be the casefor mode A, whereas for mode B, the sign of x-vorticity does not change with t → t + T/2 andy → −y.

Previous Floquet analyses have been based on the Poincare map, and for these synchronousmodes the multipliers have been reported as passing through the unit circle at µP = +1. How-ever, only modes A (wakes) and B (cavity) are H-invariant. Under the half-period-flip map H,only these modes preserve their shape and sign. The other synchronous modes preserve shapebut change sign. Floquet analysis based on the half-period-flip map shows that the H-invariantmodes (A: wakes; B: cavity) have µH = +1, while the synchronous modes that break H-symmetry (B: wakes; A: cavity) have µH = −1 (giving µP = µ2

H= +1). Note that µH = −1

is not a period doubling bifurcation for the flow φ (although it is a period doubling bifurcationfor the map H, whose period is half the period of φ), but an H-symmetry breaking bifurcation.

Figure 4 showns the Floquet multipliers for the two-dimensional wake of a circular cylinder,computed at Re = 280. At this Reynolds number, the two-dimensional basic state is unstable toboth modes A and B, while there is an intermediate-wavenumber mode (or modes) with com-plex-conjugate pair Floquet multipliers to which the basic state is stable, i.e. |µ| < 1 (Barkley& Henderson 1996). We have computed and plotted in figure 4 (a) the absolute values of theFloquet multipliers for the linearized Poincare map, |µP |, as functions of wavenumber β, and

† A half wavelength, λ/2, translation in the spanwise direction also changes the sign of thevelocity perturbation (for both modes A and B), Rλ/2u

′(x, t) = −u′(x, t), and so there is

still a spatio-temporal symmetry that leaves the wake mode B and the cavity mode A invariant:Rλ/2Hu

′(x, t) = (u′,−v′, w′)(x,−y, z + λ/2, t + T/2) = u′(x, t).


(a)

Mode B

Mode A

−1 +1Re(µH)

Im(µH)

−1

+1(b)

β

0

5

10

===

==

zzzz

zz

FIGURE 4. Floquet multipliers for the three-dimensional instability modes of the two-dimensional wake ofa circular cylinder at Re = 280: (a), |µP |, compared to the results of Barkley & Henderson (1996) — solidcircles are for modes with real Floquet multipliers, while open circles are for quasi-periodic modes withcomplex-conjugate pair multipliers; (b), loci of µH with β for the synchronous modes (solid lines) andquasi-periodic modes (dashed lines) computed using the half-period-flip map.

comparison is made to values digitized from figure 7, Barkley & Henderson (1996) — it can beseen that the agreement is quite satisfactory. In figure 4 (b), the loci of complex values of µH arepresented as a perspective view, showing that while µH for mode A are real and positive, thosefor mode B are real and negative.

At β = 0, there is always a two-dimensional marginally stable mode, with µP = µH = +1,representing the local acceleration field of the base flow at t = t0 (Guckenheimer & Holmes1986). This mode is H-symmetric. An interesting detail is that for the wake flows, mode A isH-symmetric and the locus of its multipliers connects smoothly with the two-dimensional mode(as can be seen in figure 4). For the periodically driven cavity, however, mode A breaks theH-symmetry and the connection does not occur (Blackburn & Lopez 2003b).

5. Quasi-periodic modesThe modes with complex-conjugate pair Floquet multipliers (at marginal stability, µP = e±iθ)

introduce a secondary period into the flow; the basic state loses stability through a Neimark–Sacker bifurcation. The new period Ts is related to the phase angle θ (this angle is not necessarilythe self-rotation number, see Lopez & Marques 2000; Blackburn 2002). Again, µP = µ2

H, sothat at marginal stability µH = e±iθ/2. With complex-conjugate pair multipliers, the expansions(3.4) and (3.5) for u′ are coupled (by the imaginary part of the multipliers), and the symmetriesof the problem force the multipliers to have multiplicity two. Hence, for a generic (i.e. non-resonant) Neimark–Sacker bifurcation there are four multipliers crossing the unit circle (twopairs of complex-conjugate multipliers) and so the centre manifold is four dimensional. In thiscase one cannot in general take either (3.4) or (3.5), or even a specific linear combination ofthe two, for u′. To do so corresponds to fixing the phase in z and hence imposes the symmetryof a (modulated) standing wave; modulated by the time-periodicity of the basic state. When themultipliers are complex-conjugate, a general linear combination of (3.4) and (3.5) must be usedto allow for (modulated) travelling wave solutions.

Describing the four-dimensional centre manifold in terms of the complex amplitudes of thetwo pairs of eigenmodes, (A, A) and (B, B), and writing these in polar form (A = r1 exp iφ1,


ε < 0 (I) ε > 0 ε < 0 (II) ε > 0

r1

r2

FIGURE 5. Phase portraits for the normal form (5.1), corresponding to the cases 0 < a < b (I) and0 < |b| < a (II) in the normal form. The fixed point at the origin is the basic state, the fixed points onthe horizontal (r1) axis and the vertical (r2) axis are TW, and the fixed point on the bisector is SW. Solid(hollow) points are stable (unstable).

B = r2 exp iφ2), it can be shown that the phase dynamics decouple from the dynamics ofthe amplitudes r1 and r2, and the resulting normal form is effectively a codimension-one two-dimensional map of the form (Marques et al. 2003):

r1 7→ r1(1 + ε − ar21 − br2

2), r2 7→ r2(1 + ε − ar22 − br2

1). (5.1)

This is the normal form, to third order, for both the Poincare map,P , and the half-period-flip map,H. The normal form coefficients a and b, and the bifurcation parameter ε, of P , are simply twicethose of H. This normal form admits four different fixed points, the origin (r1 = 0, r2 = 0) cor-responding to the T -periodic basic state, fixed points on each axis, ([ε/a]1/2, 0) and (0, [ε/a]1/2)corresponding to a pair of modulated travelling waves (TW), and ([ε/(a+ b)]1/2, [ε/(a+ b)]1/2)a modulated standing wave (SW). There are direct analogies between the Neimark–Sacker bi-furcation with O(2) symmetry considered here and the Hopf bifurcation with O(2) (Golubitsky,Stewart & Schaeffer 1988; Crawford & Knobloch 1991), however resonances can introduce newphenomena. Different phase portraits for (5.1) are possible, depending on the values of a andb. The two different phase portraits that occur in the problems considered here correspond to0 < a < b and 0 < |b| < a, illustrated in figure 5 as cases I and II respectively. They differ inthe stability of the bifurcated solutions. In case I the bifurcated TW are stable, while the SW isunstable; case II is the opposite.

The phase portraits shown in figure 5 are in terms of r1 and r2 only, yet the centre manifold isfour-dimensional and so we need to also consider the phases φ1 and φ2. The phase informationis recovered by independent rotations about each of the two axes, r1 and r2. In doing so, theorigin (the base state) remains a fixed point, but the two TW on the axis become circles and theSW on the bisector becomes a two-torus. The structure of these phase portraits is the same forthe Poincare and half-period-flip maps. In the continuous system, there is the additional periodof the T -periodic base state, so the origin corresponds to a limit cycle (T1), the two TW aretwo-tori (T2), and the SW a three-torus (T3). But for SW, φ1 = φ2, and so in fact there is acontinuous family of T

2 SW that span a T3; the individual SW are distinguished by their phase

in the spanwise z-direction.The flow in the 2:1 aspect ratio periodically driven cavity provides a convenient example for

the discussion of behaviours and symmetries of the modulated standing and travelling waves.With suitable choices of Re and St, mode QP (quasi-periodic) bifurcates directly from the two-dimensional base flow via a Neimark–Sacker bifurcation. The bifurcation parameter ε in (5.1)is (linearly) related to both Re and St and ε = 0 corresponds to the curve of marginal stability.Any one-parameter path in (Re, St) that crosses the curve transversally will exhibit dynamicsdescribed by (5.1). For 87.5 < St < 132, the quasi-periodic mode is the first to bifurcate as Reis increased, corresponding to ε changing from negative to positive. The new secondary period,Ts, introduced by this Neimark–Sacker bifurcation is greater than the primary period T (the walloscillation period) by a factor between four and five, depending on St (at the bifurcation point,


TWz

y

//

0 T/6 2T/6 3T/6 4T/6 5T/6 T

SWz

y

//

FIGURE 6. Vorticity dynamics of modulated standing and travelling waves in a periodically driven cavityflow (Blackburn & Lopez 2003b), shown in a domain length of one wavelength, at (St = 100, Re = 1225).Solid isosurfaces are of the out-of-page component of vorticity, positive and negative of equal magnitude,while translucent isosurfaces represent the spanwise (horizontal) component of vorticity. The driven cavitywall lies further into the page than the structures, and oscillates vertically, parallel to the page. The upperpanels, labelled TW, illustrate a modulated +z-travelling wave; the lower panels, labelled SW, illustratethe modulated standing wave.

Ts = 2πT/θ), and the wavelength λ of the instability is approximately three-quarters of thecavity width, h, in the x direction (Blackburn & Lopez 2003b). At ε = 0, both TW and SWbifurcate from the basic state supercritically, and the amplitude of TW are greater than that ofSW, and so the TW are found to be stable; this is the scenario I shown in figure 5. For TW,the wave speed s = 2π/(βTs) is approximately h/6T ; at the bifurcation point, s = θ/(βT ).Figure 6 shows the time evolution of TW and SW over one wall oscillation period T , visualizedby isosurfaces of the x and z components of vorticity.

Figure 6 shows a single spanwise wavelength, λ, of the spatially-periodic instabilities. Theinstabilities (‘braids’) of the quasi-periodic mode appear to be of centrifugal type, and grow onthe (two) main spanwise ‘rollers’, which in the view shown are oriented horizontally, and locatedtowards the upper and lower sides of the cavity. In the TW case, the braids that form on theupper and lower rollers are displaced in span by λ/4. The mechanism of wave progression isby merging of braid vortices which have like sign that are produced on opposite rollers — thisprocess can be observed in the upper panels of figure 6, which illustrates a +z-TW. A similarmerging process can be observed in the lower panels of figure 6 for SW, but owing to theirsymmetry — a superposition of ±z-TW with the same weight — no spanwise motion occurs.Figure 6 further illustrates that TW is T -periodic in a reference frame moving at wave speed,while SW is quasi-periodic.

The two TW break the Kz symmetry, however a spanwise reflection transforms one into theother. They lie on two T

2, invariant by time evolution and by spanwise translation, i.e. a transla-tion of sT/2 in z (i.e. the action of RsT/2, where s is the wave speed) is equivalent to advancing


time by T . The action of H is also equivalent to the same z translation:

Hu′(x, t) = (u′,−v′, w′)(x,−y, z, t + T/2)

= (u′, v′, w′)(x, y, z + sT/2, t) = RsT/2u′(x, t). (5.2)

So, although the TWs are not H-invariant, they are invariant to the spatio-temporal symmetryR−sT/2H . In other words, when a modulated travelling wave is observed in a reference framemoving steadily at wave speed s, it is seen as a limit cycle with spatio-temporal-symmetry H ,rather than a quasi-periodic state as it is observed in a stationary frame. This can be seen infigure 6, by comparing the frames at 0 and T . Also note that the frame at t = 0, reflected in yand translated sT/2 in z, results in the frame at T/2.

The SW, each being a superposition of waves of equal amplitude travelling in both ±z di-rections, are spanwise Z2-invariant about their nodes in z (see figure 6). These solutions arequasi-periodic in any frame of reference, and there is a complete circle of these parameterizedby their phase in z, so they span a three-torus. Translations in z (the action of the SO(2) com-ponent of the spanwise O(2) symmetry) transform one of these quasi-periodic solutions intoanother quasi-periodic solution on the same three-torus. The action of H leaves the three-torusinvariant. This is a generic feature of Z2-equivariant Neimark–Sacker bifurcations (Kuznetsov1998). But individual SW are not H-invariant, they are transformed into another SW on the samethree-torus, i.e. are shifted in span.

In summary, TW break (spanwise) Z2, retain SO(2) symmetry in z, and come in Z2-conjugatepairs. The SW break SO(2) and retain Z2 symmetry, and there is a continuous circle of these,parameterized by their phase in z through the action of SO(2). Neither SW nor TW are H-invariant, but TW are invariant to the spatio-temporal symmetry R−sT/2H , while SW are notinvariant to any spatio-temporal symmetry. The symmetry properties of the quasi-periodic modesof the circular and square-section cylinder wakes have been investigated and are identical to thoseof the periodically driven cavity flow.

In closing, we note that a true subharmonic mode, µP = −1, requires µH = e±iπ/2. Sincethe centre manifold is at least four dimensional, the theory for the suppression of period dou-bling with a simple Floquet multiplier at −1 (Swift & Wiesenfeld 1984) does not apply — in thetransition from two-dimensional to three-dimensional flow, the breaking of the spanwise O(2)symmetry gives multipliers with multiplicity two. While possible, it appears that to date no truesubharmonic mode has been reported for these flows (Blackburn & Lopez 2003a).

6. ConclusionsWe have demonstrated how Floquet analysis based on the half-period-flip map provides a

direct method to investigate symmetry breaking instabilities in spatio-temporal Z2-symmetricflows. For such flows with spanwise O(2) symmetry, we have explored the implications of thecomplete spatio-temporal symmetry, Z2 × O(2), on their transitions from two-dimensional tothree-dimensional flows in a well-studied autonomous system (periodically shedding two-dimen-sional wakes) and in a non-autonomous system (the periodically driven cavity). The symmetryimplications for quasi-periodic modes, which may be either modulated standing or modulatedtravelling waves, provides new insight into the transition process to three-dimensional flow.

This work was supported by the Australian Partnership for Advanced Computing’s Merit Al-location Scheme, the Australian Academy of Science’s International Scientific CollaborationsProgram, MCYT grant BFM2001-2350 (Spain), and NSF Grant CTS-9908599 (USA).

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on three-dimensional instabilities of two-dimensional ﬂows

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