O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

Cliffe, K. A.; Spence, A.; Tavener, Simon

doi:10.1002/(sici)1097-0363(20000130)32:2<175::aid-fld912>3.0.co;2-5

Cited by 10 publications

(10 citation statements)

References 30 publications

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“…For x = 10, the changes of Reel and Rec 2 with S follow the same trend but are weaker, increasing until Reel ｾ＠ 134 (Rec 2 ｾ＠ 148) at S = 0.5 and then decreasing to rv126 (146) at S = 0.666. Tavener (1994) and Cliffe et al (2000) observed the same trend in the case of a sphere embedded in a Poiseuille flow: Reel first increases and then decreases when the confinement ratio increases. Zovatto & Pedrizzetti (2001 ) related this effect to the interaction of the vorticity produced at the surface of the body with the vorticity produced at the wall, which have opposite signs.…”

Section: Nature Of the Path Of A Body Falling In A Tubementioning

confidence: 53%

“…Below the threshold of wake instability, Tavener (1994) and Maheshwari, Chhabra & Biswas (2006) have shown that the length of the recirculating wake of a fixed sphere decreases when the confinement ratio is increased. Numerical investigations for fixed two-dimensional cylinders (Chen, Pritchard & Tavener 1995 ;Sahin & Owens 2004) and fixed spheres (Tavener 1994;Cliffe, Spence & Tavener 2000) placed in an incoming confined flow indicate that for S < 0.5, wake instability is delayed. For 0.5 < S < 0.7, the threshold for wake instability decreases but remains larger than its value in the unconfined case.…”

Section: Introductionmentioning

confidence: 99%

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The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

Brosse¹,

Ern²

2013

J. Fluid Mech.

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To cite this version:Nicolas Brosse, Patricia Ern. The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2013, vol. 714, pp. 238-257. <10.1017/jfm.2012 This study concems the rectilinear and periodic paths of an axisymmetric solid body (short-length cylinder and disk of diameter d and thickness h) falling in a vertical tube of diameter D. We investigated experimentally the influence of the confinement ratio (S = djD < 0.8) on the motion of the body, for different aspect ratios (X = djh = 3, 6 and 1 0), Reynolds numbers (80 < Re < 320) and a density ratio between the fluid and the body close to unity. For a given body, the Reynolds number based on its mean vertical velocity is observed to decrease when S increases. The critical Reynolds number for the onset of the periodic motion decreases with S in the case of thin bodies (X = 10), whereas it appears unaffected by S for thicker bodies (X = 3 and 6). The characteristics of the periodic motion are also strongly modified by the confinement ratio. A thick body (x = 3) tends to go back to a rectilinear path when S increases, while a thin body (x = 1 0) displays oscillations of growing amplitude with S un til it touches the tube (at about S = 0.5). For a given aspect ratio, however, the amplitudes of the oscillations follow a unique curve for all S, which depends only on the relative distance of the Reynolds number to the threshold of path instability. In parallel, numerical simulations of the wake of a body held fixed in a uniform confined flow were carried out. The simulations allowed us to determine in this configuration the effect of the confinement ratio on the thresholds for wake instability (loss of axial symmetry at Rec1 and loss of stationarity at Rec 2 ) and on the maximal velocity Vw in the recirculating region of the stationary axisymmetric wake. The evolution with x and S of Vw at Rec1 was used to define a Reynolds number Re*. Remarkably, for a freely moving body, Re* remains almost constant when S varies, regardless of the nature of the path.

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Section: Nature Of the Path Of A Body Falling In A Tubementioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

Brosse¹,

Ern²

2013

J. Fluid Mech.

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“…They treat finite symmetry groups. Cliffe et al [2] developed numerical methods for the computation of bifurcations of stationary solutions in the case of continuous rotational symmetries.…”

Section: Introductionmentioning

confidence: 99%

Numerical Continuation of Symmetric Periodic Orbits

Wulff¹,

Schebesch²

2006

SIAM J. Appl. Dyn. Syst.

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The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for symmetric dynamical systems. But there are hardly any results on the numerical computation of those bifurcations yet. In this paper we show how spatiotemporal symmetries of periodic orbits can be exploited numerically. We describe methods for the computation of symmetry breaking bifurcations of periodic orbits for free group actions and show how bifurcations increasing the spatiotemporal symmetry of periodic orbits (including period halving bifurcations and equivariant Hopf bifurcations) can be detected and computed numerically. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a a tangential continuation method with implicit reparametrization.

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“…In this article, we study the stability of the three-dimensional incompressible Navier-Stokes equations in the case when the underlying system possesses both rotational and reflectional symmetry, or more precisely, O(2) symmetry. To this end, we are interested in numerically estimating the critical Reynolds number Re, at which a (pitchfork) bifurcation point first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al [13], for example. The work in this article expands upon our recent work in [12] and [10] to include problems whose geometries exhibit both rotational and reflectional symmetry.…”

mentioning

confidence: 99%

“…See Cliffe et al [13]. The decomposition (3.4) is produced purely by the rotation elements r α of O(2) and the fact that there is also a reflection element has not been used.…”

mentioning

confidence: 99%

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

et al. 2011

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Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1437/1/cliffe_et_al_o2_article.pdf Copyright and reuse:The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses rotational and reflectional or O(2) symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual Weighted Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented. Here, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.Key words. Incompressible flows, bifurcation problems, a posteriori error estimation, adaptivity, discontinuous Galerkin methods, O(2) symmetry 1. Introduction. In this article, we study the stability of the three-dimensional incompressible Navier-Stokes equations in the case when the underlying system possesses both rotational and reflectional symmetry, or more precisely, O(2) symmetry. To this end, we are interested in numerically estimating the critical Reynolds number Re, at which a (pitchfork) bifurcation point first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al. [13], for example. The work in this article expands upon our recent work in [12] and [10] to include problems whose geometries exhibit both rotational and reflectional symmetry. The detection of bifurcation points in this setting is now well understood, for example, see Golubitsky and Schaeffer [19]. For the purposes of this article, we assume that a symmetric steady state solution to the incompressible Navier-Stokes equations undergoes a steady pitchfork bifurcation at a critical value of the Reynolds number. Estimation of the critical...

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O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

Cited by 10 publications

References 30 publications

The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

Numerical Continuation of Symmetric Periodic Orbits

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

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