On stability of Taylor vortices by fifth-order amplitude expansions

Eagles, P. M.

doi:10.1017/s0022112071002246

Cited by 72 publications

(39 citation statements)

References 7 publications

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“…We follow Eagles [5] and write the disturbance equations in dimensionless matrix form as follows. We set u = u', v = v s + v', w = w s + w', p =Ps + P' and ignore terms of O(e4).…”

Section: The Perturbation Equations In Matrix Formmentioning

confidence: 99%

“…We need to use the adjoint eigenfunction as defined in Eagles [5]. This is denoted by f~ where aL ~---~ + { A~') }trL = O.…”

Section: Solutions Of the Disturbance Equationsmentioning

confidence: 99%

“…The methods of computation are similar to those of Eagles [5] and need not be described again, except to say that numerous consistency checks were made by changing arbitrarily the scalings of the various functions involved and following through the results numerically and theoretically. Comparisons were made with earlier work wherever possible.…”

Section: The Solution For S(z*)mentioning

confidence: 99%

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Taylor vortices between almost cylindrical boundaries

Eagles

Eames

1983

J Eng Math

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We consider a modified Taylor problem, with the fluid flowing between a rotating inner circular cylinder and a n outer stationary surface whose radius is a constant plus a small and slowly varying function of the axial co-ordinate z. This variation is chosen in such a way that the flow is locally more unstable near z = 0 than near z = _+ oo, so that Taylor vortices appear more readily near z = 0. The theory is developed to show how vortices of strength varying with z develop as the speed of rotation is increased through a critical value which is a perturbation of the classical value. Wave number changes in the axial direction are also calculated.

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“…We follow Eagles [5] and write the disturbance equations in dimensionless matrix form as follows. We set u = u', v = v s + v', w = w s + w', p =Ps + P' and ignore terms of O(e4).…”

Section: The Perturbation Equations In Matrix Formmentioning

confidence: 99%

“…We need to use the adjoint eigenfunction as defined in Eagles [5]. This is denoted by f~ where aL ~---~ + { A~') }trL = O.…”

Section: Solutions Of the Disturbance Equationsmentioning

confidence: 99%

Section: The Solution For S(z*)mentioning

confidence: 99%

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Taylor vortices between almost cylindrical boundaries

Eagles

Eames

1983

J Eng Math

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“…The Taylor vortices were shown to develop due to a linear inviscid instability of the base flow, putting experiment and theory in good agreement with one another and providing some hope that the phenomena of transition and turbulence may be theoretically understood. Taylor's experiments sparked a flurry of research activity, which includes the following: theoretical studies of Taylor vortex instability such as Davey et al (1968) and Eagles (1971), who used weakly nonlinear analysis to determine the instabilities which affect Taylor vortices; experimental papers such as Coles (1965), Andereck et al (1986) and Hegseth et al (1996), who mapped in parameter space the different flow regimes observed in the experiments; and numerical stability analyses wherein finite-amplitude Taylor vortices are calculated numerically in addition to the higher-order structures they bifurcate towards, such as Nagata (1988), Weisshaar et al (1991) and Antonijoan & Sánchez (2000). Recent experimental attention has been focused on the flow of a differentially rotated fluid through a linear shear layer, known as rotating plane Couette flow (RPCF), a review of which can be found in Mullin (2010).…”

Section: Introductionmentioning

confidence: 99%

Secondary instability and tertiary states in rotating plane Couette flow

et al. 2014

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Recent experimental studies have shown rich transition behaviour in rotating plane Couette flow (RPCF). In this paper we study the transition in supercritical RPCF theoretically by determination of equilibrium and periodic orbit tertiary states via Floquet analysis on secondary Taylor vortex solutions. Two new tertiary states are discovered which we name oscillatory wavy vortex flow (oWVF) and skewed vortex flow (SVF). We present the bifurcation routes and stability properties of these new tertiary states, and in addition, we describe a bifurcation procedure whereby a set of defected wavy twist vortices are approached. Further to this, transition scenarios at flow parameters relevant to experimental works are investigated by computation of the set of stable attractors which exist on a large domain. The physically observed flow states are shown to share features with states in our set of attractors.

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“…They computed the cubic coefficients in their model (without exploiting symmetry to simplify the calculations). The fifth-order coefficients were obtained by Eagles (1971). The reduction to a 6-dimensional model was not justified rigorously; indeed they worked with a fixed outer cylinder, where the requisite mode interaction does not actually occur.…”

Section: A N D R E W H I L L a N D I A N S T E W A R Tmentioning

confidence: 99%

3-mode Interactions with O(2) Symmetry and a Model for Taylor-Couette flow

Hill

Stewart

1991

Dynamics and Stability of Systems

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We analyse the bifurcations of a general ordinary differential equationwhere fRIO x R2 x R + RI O is equivariant under an action of the group O(2) on R'O. The equation represents the most general nonlinear local interaction of three O(2)-symmetric modes: a steady-state mode with mode-number k, and two periodic (Hopf) modes with mode-numbers 1 and m. The parameter I. is a bifurcation parameter, and a,, a, are unfolding parameters that split the individual modes apart. The system is assumed to be in Birkhoff normal form, so that f also commutes with an action of the 2-torus T2. We discuss the existence and stability of bifurcating branches and how these break the O(2) x T2 symmetry.Depending on the precise mode-numbers k, 1, m we find up to 31 symmetry classes of possible solutions including six that combine all three modes, and thus cannot be found in any 2-mode interaction. We also discuss the possible occurrence of Sacher-Naimark torus bifurcations, providing a further 10 solution types, and 'slow drift' bifurcations.This 10-dimensional system can occur generically in O(2)-symmetric bifurcation problems having two extra parameters, and in principle is applicable to a wide range of physical systems. The discussion here is motivated by the observed pattern formation in the Taylor-Couette system, the flow of a fluid contained between coaxial rotating cylinders. It arises by seeking a 'hidden organizing centre' that combines two previous mode-interaction models of this system: a 6-dimensional Hopf-steady-state model due to Chossat and Iooss (1985) and Golubitsky and Stewart (1986), and an 8-dimensional Hopf-Hopf model due to Chossat, Demay and Iooss (1987). We interpret the general results on the 10-dimensional system in the context of Taylor-Couette flow, giving schematic pictures of the associated flow patterns. The model incorporates almost all of the observed non-chaotic flows in the Taylor-Couette experiment into a single finite-dimensional dynamical system. It predicts the possible occurrence of four new flow patterns (corresponding to four of the six possible solutions that combine all three modes). These form invariant 3-tori, and may be described as superimposed twisted vortices, superimposed wavy vortices, and two types of twisted wavy vortices. Possible torus bifurcations from states in the 10-dimensional model include various modulated spirals, three types of modulated twisted vortices, three types of modulated wavy vortices, modulated superimposed spirals, modulated interpenetrating spirals, modulated superimposed ribbons, and modulated interpenetrating ribbons. However, whether any of these new states and torus bifurcations can actually occur in Taylor-Couette flow at suitable parameter values, and if so whether they can occur stably, depend upon more detailed numerical calculations than we have performed. 0 Oxford University Press 1991

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On stability of Taylor vortices by fifth-order amplitude expansions

Cited by 72 publications

References 7 publications

Taylor vortices between almost cylindrical boundaries

Taylor vortices between almost cylindrical boundaries

Secondary instability and tertiary states in rotating plane Couette flow

3-mode Interactions with O(2) Symmetry and a Model for Taylor-Couette flow

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