Floquet stability analysis of the flow around an oscillating cylinder

Gioria, Rafael dos Santos; Jabardo, Paulo José Saiz; Carmo, Bruno Souza; Meneghini, Júlio Romano

doi:10.1016/j.jfluidstructs.2009.01.004

Cited by 23 publications

(14 citation statements)

References 28 publications

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“…An extra term is simply added to the right-hand side of the quations to account for the non-inertia of the reference frame. This approach is more efficient than employing a deforming mesh to tackle the disk oscillation, as has been used in previous studies [25][26][27]. This renders the system to be solved as…”

Section: B Numerical Methodologymentioning

confidence: 99%

Instabilities of interacting vortex rings generated by an oscillating disk

et al. 2016

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We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c, undergoing oscillations of frequency f_{0} and amplitude A. We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly on the azimuthal (integer) wave number m. We vary two key control parameters, the Keulegan-Carpenter number K_{C}=2πA/c and the Stokes number β=f_{0}c^{2}/ν, where ν is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small β, and hence low frequency of oscillation corresponding to relatively weak interaction between sequentially shedding vortex rings, symmetry breaking occurs directly to a single unstable mode with m=1. Second, for sufficiently large yet fixed values of β, corresponding to a higher oscillation frequency and hence stronger ring-ring interaction, the onset of asymmetry is predicted to occur due to two branches of high m instabilities as the amplitude is increased, with m=1 structures being dominant only for sufficiently large values of K_{C}. These two branches can be distinguished by the phase properties of the vortical structures above and below the disk. The region in (K_{C},β) parameter space where these two high m instability branches arise can be described accurately in terms of naturally defined Reynolds numbers, using appropriately chosen characteristic length scales. We subsequently carry out direct numerical simulations of the fully three-dimensional flow to verify the principal characteristics of the Floquet analysis, in particular demonstrating that high wave-number symmetry-breaking generically occurs when vortex rings sequentially interact sufficiently strongly.

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Section: B Numerical Methodologymentioning

confidence: 99%

Instabilities of interacting vortex rings generated by an oscillating disk

et al. 2016

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“…It is noted that a real harmonic mode cannot be physically realized unless the Z2 spatiotemporal symmetry in the wake is broken. This can be achieved in two ways: (i) in geometries without reflection symmetry, such as a circular cylinder with a tripwire placed adjacent to the cylinder but not on the symmetry plane [12], inclined flat plates [13], inclined square cylinders [14], and stalled airfoils [10]; (ii) by a transversely oscillating cylinder which can also change the spatiotemporal symmetry of the two-dimensional wake [15,16], At high oscillation amplitudes, the wake takes on the "P + S" configuration, with a pair of vortices on one side ol the wake and a single vortex on the other side for each oscillation cycle. As a result of the asymmetry about the center line, a real subharmonic "mode C" instability emerges, or more specifically for oscillating cylinders two subharmonic modes, "S L " and "55'," appear, with long and short wavelengths respectively.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional transition after wake deflection behind a flapping foil

Deng

Caulfield

2015

Phys. Rev. E

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We report the inherently three-dimensional linear instabilities of a propulsive wake, produced by a flapping foil, mimicking the caudal fin of a fish or the wing of a flying animal. For the base flow, three sequential wake patterns appear as we increase the flapping amplitude: Bénard-von Kármán (BvK) vortex streets; reverse BvK vortex streets; and deflected wakes. Imposing a three-dimensional spanwise periodic perturbation, we find that the resulting Floquet multiplier |μ| indicates an unstable "short wavelength" mode at wave number β=30, or wavelength λ=0.21 (nondimensionalized by the chord length) at sufficiently high flow Reynolds number Re=Uc/ν≃600, where U is the upstream flow velocity, c is the chord length, and ν is the kinematic viscosity of the fluid. Another, "long wavelength" mode at β=6 (λ=1.05) becomes critical at somewhat higher Reynolds number, although we do not expect that this mode would be observed physically because its growth rate is always less than the short wavelength mode, at least for the parameters we have considered. The long wavelength mode has certain similarities with the so-called mode A in the drag wake of a fixed bluff body, while the short wavelength mode appears to have a period of the order of twice that of the base flow, in that its structure seems to repeat approximately only every second cycle of the base flow. Whether it is appropriate to classify this mode as a truly subharmonic mode or as a quasiperiodic mode is still an open question however, worthy of a detailed parametric study with various flapping amplitudes and frequencies.

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“…On the other hand, if at least one of the Floquet multipliers has a modulus greater than one, then that perturbation will grow exponentially and the orbit is unstable; see, e.g. Barkley & Henderson (1996) and Gioria et al (2009). In our study the Floquet exponents are complex and the imaginary part ω F of the Floquet exponent is the argument (angle) of the Floquet multiplier.…”

Section: Mean Flow and Linearizationmentioning

confidence: 76%

“…Barkley & Henderson (1996) and Gioria et al. (2009). In our study the Floquet exponents are complex and the imaginary part of the Floquet exponent is the argument (angle) of the Floquet multiplier.…”

Section: Governing Equations and Numerical Methodsmentioning

confidence: 99%

Bifurcation analysis and frequency prediction in shear-driven cavity flow

et al. 2019

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A comprehensive study of the two-dimensional incompressible shear-driven flow in an open square cavity is carried out. Two successive bifurcations lead to two limit cycles with different frequencies and different numbers of structures which propagate along the top of the cavity and circulate in its interior. A branch of quasi-periodic states produced by secondary Hopf bifurcations transfers the stability from one limit cycle to the other. A full analysis of this scenario is obtained by means of nonlinear simulations, linear stability analysis and Floquet analysis. We characterize the temporal behaviour of the limit cycles and quasi-periodic state via Fourier transforms and their spatial behaviour via the Hilbert transform. We address the relevance of linearization about the mean flow. Although here the nonlinear frequencies are not very far from those obtained by linearization about the base flow, the difference is substantially reduced when eigenvalues are obtained instead from linearization about the mean and in addition, the corresponding growth rate is small, a combination of properties called RZIF (real zero imaginary frequency). Moreover growth rates obtained by linearization about the mean of one limit cycle are correlated with relative stability to the other limit cycle. Finally, we show that the frequencies of the successive modes are separated by a constant increment.

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Floquet stability analysis of the flow around an oscillating cylinder

Cited by 23 publications

References 28 publications

Instabilities of interacting vortex rings generated by an oscillating disk

Instabilities of interacting vortex rings generated by an oscillating disk

Three-dimensional transition after wake deflection behind a flapping foil

Bifurcation analysis and frequency prediction in shear-driven cavity flow

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