Bifurcations in systems with Z2 spatio-temporal and O(2) spatial symmetry

Marqués, Francisco; Lopez, Juan M.; Blackburn, Hugh Maurice

doi:10.1016/j.physd.2003.09.041

Cited by 50 publications

(89 citation statements)

References 22 publications

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“…The standing wave SW does not have oblique streamwise vortices, and is K z invariant. Let us compare these results with the normal form analysis for the codimension-one bifurcation with complex eigenvalues, F c 4 , for systems with spatial symmetry O(2) and spatio-temporal symmetry Z 2 (Marques et al 2004). In the F c 4 bifurcation, there is a pair of complex-conjugate Floquet multipliers, µ H = e ±iθ/2 , θ ∈ (0, 2π), of multiplicity two (i.e.…”

Section: Circular Cylinder Wakesmentioning

confidence: 99%

Symmetry breaking of two-dimensional time-periodic wakes

2005

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A number of two-dimensional time-periodic flows, for example the Kármán street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the solution invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincaré map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-periodflip map provides a comprehensive interpretation of the symmetry breaking bifurcations.

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Section: Circular Cylinder Wakesmentioning

confidence: 99%

Symmetry breaking of two-dimensional time-periodic wakes

2005

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“…In contrast, autonomous time-periodic flows, such as two-dimensional circular and square cylinder wakes, typically possess only one control parameter (the free-stream velocity as characterized by the Reynolds number), and consequently, the three modes bifurcate from the base state sequentially as this single control parameter is varied. explored the relationship between the threedimensional instabilities of the idealized two-dimensional cylinder wake and driven cavity flows, and Marques, Lopez & Blackburn (2004) provided a detailed centre manifold and normal form analysis of the bifurcations involved. An experimental test of the theoretical predictions from Marques et al (2004) was presented in Leung et al (2005), where the physical periodically driven cavity problem with finite span had Z 2 × Z 2 spatio-temporal symmetry (the spanwise translation/reflection invariance O(2) in the model problem being reduced to a spanwise reflection Z 2 in the physical experiment).…”

Section: Introductionmentioning

confidence: 99%

Modulated waves in a periodically driven annular cavity

Blackburn

Lopez

2010

J. Fluid Mech.

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Time-periodic flows with spatio-temporal symmetry Z 2 × O(2) -invariance in the spanwise direction generating the O(2) symmetry group and a half-period-reflection symmetry in the streamwise direction generating a spatio-temporal Z 2 symmetry group -are of interest largely because this is the symmetry group of periodic laminar two-dimensional wakes of symmetric bodies. Such flows are the base states for various three-dimensional instabilities; the periodically shedding two-dimensional circular cylinder wake with three-dimensional modes A and B being the generic example. However, it is not easy to physically realize the ideal flows owing to the presence of end effects and finite spanwise geometries. Flows past rings are sometimes advanced as providing a relevant idealization, but in fact these have symmetry group O(2) and only approach Z 2 × O(2) symmetry in the infinite aspect ratio limit. The present work examines physically realizable periodically driven annular cavity flows that possess Z 2 × O(2) spatio-temporal symmetry. The flows have three distinct codimension-1 instabilities: two synchronous modes (A and B), and two manifestations of a quasi-periodic (QP) mode, either as modulated standing waves or modulated travelling waves. It is found that the curvature of the system can determine which of these modes is the first to become unstable with increasing Reynolds number, and that even in the nonlinear regime near onset of three-dimensional instabilities the dynamics are dominated by mixed modes with complicated spatio-temporal structure. Supplementary movies illustrating the spatio-temporal dynamics are available at journals.cambridge.org/flm.

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“…The synchronous bifurcations are pitchforks of revolution that can break (mode A) or preserve (mode B) the space-time symmetry H . The Neimark-Sacker bifurcations result in quasiperiodic solutions that are modulated traveling or standing waves [11]. Figure 2 Three-dimensional synchronous states appear when a pair of real eigenvalues cross the unit circle at +1 in the complex plane.…”

Section: A Previous Resultsmentioning

confidence: 99%

“…Furthermore, these flows have an additional space-time symmetry: a reflection about the wake centerline followed by a half-period temporal evolution. In several wake flows, two distinct synchronous modes that break into the spanwise direction (with real Floquet exponent) have been observed experimentally [1][2][3][4], computed as direct instabilities from the flow [2,[5][6][7][8], and studied theoretically [9][10][11]. These modes are associated with breaking or preserving the spatiotemporal symmetry.…”

Section: Introductionmentioning

confidence: 99%

“…These problems possess more governing parameters, and by a convenient selection of them, the different modes appear at the first bifurcation. One of these problems is the flow in a rectangular cavity driven by the periodic oscillation of one wall [11][12][13][14][15]. In this case, both the synchronous and the quasiperiodic modes have been obtained and analyzed experimentally and numerically as the primary bifurcation for different control parameters.…”

Section: Introductionmentioning

confidence: 99%

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Mode competition in cylindrical flows driven by sidewall oscillations

2013

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The transition from a two-dimensional to three-dimensional flow in systems with spatial O(2) symmetry and spatiotemporal Z 2 symmetry happens in many fluid systems, like wakes or periodically forced flows. In most of these systems, the dynamics after the first bifurcation is very complex and involves cascades of bifurcations in a very narrow parameter range. A numerical study of a flow in an enclosed cylindrical cavity driven by axial oscillations of the sidewall, which allows a detailed study of the secondary bifurcations and the corresponding mode interactions, is presented. The study focuses on a codimension-2 point that acts as the organizing center of the dynamics for moderate values of the forcing frequency. The unraveled dynamics is very rich, including slow-fast dynamics and hysteresis, and may help understand the bifurcation cascades in more complex systems.

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Bifurcations in systems with Z2 spatio-temporal and O(2) spatial symmetry

Cited by 50 publications

References 22 publications

Symmetry breaking of two-dimensional time-periodic wakes

Symmetry breaking of two-dimensional time-periodic wakes

Modulated waves in a periodically driven annular cavity

Mode competition in cylindrical flows driven by sidewall oscillations

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