scholarly journals Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow

2011 ◽  
Vol 670 ◽  
pp. 96-129 ◽  
Author(s):  
F. MELLIBOVSKY ◽  
B. ECKHARDT
Keyword(s):  
Relative Equilibria ◽  
Travelling Wave ◽  
Space Structures ◽  
Periodic Flows ◽  
Parabolic Profile ◽  
Time Stepping ◽  
Higher Order Terms ◽  
Modulated Waves ◽  
Secondary Bifurcations ◽  
Unstable Branch

The appearance of travelling-wave-type solutions in pipe Poiseuille flow that are disconnected from the basic parabolic profile is numerically studied in detail. We focus on solutions in the twofold azimuthally-periodic subspace because of their special stability properties, but relate our findings to other solutions as well. Using time-stepping, an adapted Krylov–Newton method and Arnoldi iteration for the computation and stability analysis of relative equilibria, and a robust pseudo-arclength continuation scheme, we unfold a double-zero (Takens–Bogdanov) bifurcating scenario as a function of Reynolds number (Re) and wavenumber (κ). This scenario is extended, by the inclusion of higher-order terms in the normal form, to account for the appearance of supercritical modulated waves emanating from the upper branch of solutions at a degenerate Hopf bifurcation. We provide evidence that these modulated waves undergo a fold-of-cycles and compute some solutions on the unstable branch. These waves are shown to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, in accordance with the bifurcation scenario proposed. The travelling-wave upper-branch solutions are stable within the subspace of twofold periodic flows, and their subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.

scholarly journals Marangoni convection in binary mixtures with Soret effect

1998 ◽  
Vol 375 ◽  
pp. 143-177 ◽  
Author(s):  
A. BERGEON ◽  
D. HENRY ◽  
H. BENHADID ◽  
L. S. TUCKERMAN
Keyword(s):  
Marangoni Convection ◽  
Soret Effect ◽  
Spectral Element ◽  
Soret Coefficient ◽  
Small Range ◽  
Concentration Gradients ◽  
Time Stepping ◽  
Pure Fluid ◽  
Secondary Bifurcations ◽  
Steady Convection

Marangoni convection in a differentially heated binary mixture is studied numerically by continuation. The fluid is subject to the Soret effect and is contained in a two-dimensional small-aspect-ratio rectangular cavity with one undeformable free surface. Either or both of the temperature and concentration gradients may be destabilizing; all three possibilities are considered. A spectral-element time-stepping code is adapted to calculate bifurcation points and solution branches via Newton's method. Linear thresholds are compared to those obtained for a pure fluid. It is found that for large enough Soret coefficient, convection is initiated predominantly by solutal effects and leads to a single large roll. Computed bifurcation diagrams show a marked transition from a weakly convective Soret regime to a strongly convective Marangoni regime when the threshold for pure fluid thermal convection is passed. The presence of many secondary bifurcations means that the mode of convection at the onset of instability is often observed only over a small range of Marangoni number. In particular, two-roll states with up-flow at the centre succeed one-roll states via a well-defined sequence of bifurcations. When convection is oscillatory at onset, the limit cycle is quickly destroyed by a global (infinite-period) bifurcation leading to subcritical steady convection.

scholarly journals Simulation of convective transport during frequency chirping of a TAE using the MEGA code

2022 ◽  
Author(s):  
Hooman Hezaveh Hesar Maskan ◽  
Y Todo ◽  
Zhisong Qu ◽  
Boris N Breizman ◽  
Matthew J Hole
Keyword(s):  
Phase Space ◽  
Coarse Graining ◽  
Travelling Wave ◽  
Convective Transport ◽  
Space Structures ◽  
Tokamak Plasmas ◽  
Linear Evolution ◽  
Damping Rate ◽  
Space Dynamics ◽  
Frequency Chirping

Abstract We present a procedure to examine energetic particle phase-space during long range frequency chirping phenomena in tokamak plasmas. To apply the proposed method, we have performed self-consistent simulations using the MEGA code and analyzed the simulation data. We demonstrate a travelling wave in phase-space and that there exist specific slices of phase-space on which the resonant particles lie throughout the wave evolution. For non-linear evolution of an n=6 toroidicity-induced Alfven eigenmode (TAE), our results reveal the formation of coherent phase-space structures (holes/clumps) after coarse-graining of the distribution function. These structures cause a convective transport in phase-space which implies a radial drift of the resonant particles. We also demonstrate that the rate of frequency chirping increases with the TAE damping rate. Our observations of the TAE behaviour and the corresponding phase-space dynamics are consistent with the Berk-Breizman (BB) theory.

Secondary bifurcations of Taylor vortices into wavy inflow or outflow boundaries

1986 ◽  
Vol 173 ◽  
pp. 273-288 ◽  
Author(s):  
G. Iooss
Keyword(s):  
Axial Direction ◽  
Travelling Wave ◽  
Taylor Vortex ◽  
Physical Situation ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Expansion Procedure ◽  
Oscillating Solutions ◽  
Translational Mode ◽  
Secondary Bifurcations

Experiments of Andereck et al. (1986) with corotating cylinders, show that Taylor-vortex flow (TVF) can bifurcate into one of the following cellular flows: wavy vortices (WV), twisted vortices (TW), wavy inflow boundaries (WIB), wavy outflow boundaries (WOB). We describe here the structure of these different flows, showing how they result from simple symmetry breaking. Moreover we consider the codimension-two situation where WIB and WOB interact, since this is an observed physical situation.The method used in this paper is based on symmetry arguments. It differs notably from the Liapunov-Schmidt reduction used in particular by Golubitsky & Stewart (1986) on the same problem with counter-rotating cylinders. Here we take into account all the dynamics, instead of restricting the study to oscillating solutions. In addition to the standard oscillatory modes, we have a translational mode due to the indeterminacy of TVF under the shifts along the axis. We derive an amplitude-expansion procedure which allows the translational mode to depend on time. Our amplitude equations have nevertheless a simple structure because the oscillatory modes have a precise symmetry. They break, in general, the rotational invariance and they are either symmetric or antisymmetric with respect to the plane z = 0. Moreover, the most typical cases are when either of these modes has the same axial period as TVF or when their axial period is double this. This leads to four different cases which are shown to give WV, TW, WIB or WOB, all these flows being ‘rotating waves’, i.e. they are steady in a suitable rotating frame.Finally we consider the interaction between WIB and WOB that occurs when, at the onset of instability, the two critical modes arise simultaneously. In this case we show in particular that there may exist a stable quasi-periodic flow bifurcating from WIB or WOB. The two main frequencies are those of underlying WIB and WOB, while there may exist a third frequency corresponding to a slow superposed travelling wave in the axial direction.The method was used in the counter-rotating case for interacting non-axisymmetric modes (see Chossat et al. 1986). One of the original contributions here is not only to clarify the origin of all observed bifurcations from TVF, but also to handle the translational mode which may not stay small. This technique combined with centre-manifold and equivariance techniques may be helpful for many problems starting with orbits of solutions, such as the TVF considered here.

The Takens-Bogdanov bifurcation with O(2)-symmetry

1987 ◽  
Vol 322 (1565) ◽  
pp. 243-279 ◽  
Keyword(s):  
Travelling Waves ◽  
Versal Deformation ◽  
Classical Motion ◽  
Averaged Equations ◽  
Particular Solution ◽  
Pitchfork Bifurcations ◽  
Modulated Waves ◽  
Straight Lines ◽  
Secondary Bifurcations ◽  
Force Problem

The versal deformation of a vector field of co-dimension two that is equivariant under a representation of the symmetry group O(2 ) and has a nilpotent linearization at the origin is studied. An appropriate scaling allows us to formulate the problem in terms of a central-force problem with a small dissipative perturbation. We derive and analyse averaged equations for the angular momentum and the energy of the classical motion. The unfolded system possesses four different types of non-trivial solutions: a steady-state and three others, which are referred to in a wave context as travelling waves, standing waves and modulated waves. The plane of unfolding parameters is divided into a number of regions by (approximately) straight lines corresponding to primary and secondary bifurcations. Crossing one of these lines leads to the appearance or disappearance of a particular solution. We locate secondary saddlenode, Hops and pitchfork bifurcations as well as three different global, i.e. homoclinic and heteroclinic, bifurcations.

Nonlinear compressible magnetoconvection Part 2. Streaming instabilities in two dimensions

1994 ◽  
Vol 280 ◽  
pp. 227-253 ◽  
Author(s):  
M. R. E. Proctor ◽  
N. O. Weiss ◽  
D. P. Brownjohn ◽  
N. E. Hurlburt
Keyword(s):  
Numerical Experiments ◽  
Two Dimensions ◽  
Horizontal Shear ◽  
Small Aspect Ratio ◽  
Restoring Force ◽  
Dramatic Form ◽  
Modulated Waves ◽  
Secondary Bifurcations ◽  
Weak Fields ◽  
Steady Convection

We have conducted further numerical experiments on two-dimensional fully compressible convection in an imposed vertical magnetic field and interpreted the results by reference to appropriate low-order models. Here we focus on streaming instabilities, involving horizontal shear flows, which form an important mechanism for the breakdown of steady convection in relatively weak fields for boxes of sufficiently small aspect ratio. While these shearing modes can arise even in the absence of a field, they will typically lead only to travelling and modulated waves unless there is a field to provide a restoring force. For magnetoconvection a new and dramatic form of pulsating wave appears after a complex sequence of secondary bifurcations.

Third-harmonic resonance in the interaction of capillary and gravity waves

1971 ◽  
Vol 48 (2) ◽  
pp. 385-395 ◽  
Author(s):  
Ali Hasan Nayfeh
Keyword(s):  
Gravity Waves ◽  
Multiple Scales ◽  
Second Harmonic ◽  
Travelling Wave ◽  
Wave Number ◽  
Third Harmonic ◽  
Resonant Wave ◽  
Near Resonance ◽  
Modulated Waves ◽  
Harmonic Resonance

The method of multiple scales is used to determine the temporal and spatial variation of the amplitudes and phases of capillary-gravity waves in a deep liquid at or near the third-harmonic resonant wave-number. This case corresponds to a wavelength of 2·99 cm in deep water. The temporal variation shows that the motion is always bounded, and the general motion is an aperiodic travelling wave. The analysis shows that pure amplitude-modulated waves are not possible in this case contrary to the second-harmonic resonant case. Moreover, pure phase-modulated waves are periodic even near resonance because the non-linearity adjusts the phases to yield perfect resonance. These periodic waves are found to be unstable, in the sense that any disturbance would change them into aperiodic waves.

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