A spotlike edge state in plane Poiseuille flow

Zammert, Stefan; Eckhardt, Bruno

doi:10.1002/pamm.201410283

Cited by 5 publications

(9 citation statements)

References 14 publications

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“…Edge tracking calculations starting from the disturbed localised periodic orbit do not result in a simple attractor. Instead, the time evolution of the state is chaotic, but it remains localised (Zammert & Eckhardt 2014b). This behaviour is similar to what has been seen in large plane Couette domains (Marinc et al 2010;Schneider et al 2010b;Duguet et al 2009), long pipes (Mellibovsky et al 2009), or wide domains in the asymptotic suction boundary layer (Khapko et al 2014).…”

Section: A Streamwise and Spanwise Localised Periodic Orbitsupporting

confidence: 76%

Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

Zammert

Eckhardt

2014

J. Fluid Mech.

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We study localised exact coherent structures in plane Poiseuille flow that are relative periodic orbits. They are obtained from extended states in smaller periodically continued domains, by increasing the length to obtain streamwise localisation and then by increasing the width to achieve spanwise localisation. The states maintain the travelling wave structure of the extended states, which is then modulated by a localised envelope on larger scales. In the streamwise direction, the envelope shows exponential localisation, with different exponents on the upstream and downstream sides. The upstream exponent increases linearly with Reynolds number Re, but the downstream exponent is essentially independent of Re. In the spanwise direction the decay is compatible with a power-law localisation. As the width increases the localised state undergoes further bifurcations which add additional unstable directions, so that the edge state, the relative attractor on the boundary between the laminar and turbulent motions, in the system becomes chaotic.

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Section: A Streamwise and Spanwise Localised Periodic Orbitsupporting

confidence: 76%

Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

Zammert

Eckhardt

2014

J. Fluid Mech.

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“…Note that the resulting state space representation is 55-dimensional. In the numerical results to follow, we simulate this 55-dimensional ODE system (44)(45)(46)(47) using odeint function of scipy.integrate module 32 of Python programming language. Note that odeint adapts the time-steps in order to keep estimated numerical integration errors below 10 −8 .…”

Section: A Pilot-wave Model With Central Harmonic Forcementioning

confidence: 99%

“…In the results of the following sections, the trajectories are sampled with a time-step δτ = 0.01. For simulations in the symmetry-reduced state space, we integrate (19), which we obtain explicitly by projecting (44)(45)(46)(47).…”

Section: A Pilot-wave Model With Central Harmonic Forcementioning

confidence: 99%

State space geometry of the chaotic pilot-wave hydrodynamics

Budanur

Fleury²

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the formation of a global attractor. In this final regime, the droplet's angular momentum spontaneously changes its sign as observed in the experiments of Perrard et al. (Phys. Rev. Lett., 113(10):104101, 2014). Keywords: hydrodynamic quantum analogs, symmetry reductionDuring the quantum physics' infancy, Louis de Broglie 1 imagined it as the outcome of the dynamics of a point-like particle that is interacting with a continuous background field. This viewpoint was to a large extent forgotten during the second half of the twentieth century due to the great success of the Copenhagen interpretation. In recent years, a resurgent interest in de Broigle's "wave-particle duality" has developed as a result of its discovery in a completely different field: fluid mechanics. In a series of experiments pioneered by Couder et al. 2,3 , several phenomena that were once thought to be exclusive to quantum physics were demonstrated in the macroscopic setting of a droplet of silicon oil bouncing on vertically vibrating bath of the same fluid. We examine a class of these systems with rotational symmetry and formulate a novel method for their analysis. We demonstrate the utility of our approach in a numerical study where we describe the chaotic dynamics of a hydrodynamic pilotwave model through a geometrical approach, in which we identify solutions with qualitative differences and intermittent transitions in between.

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“…On a reduced-order model of a shear flow, Skufca et al [2] showed that the asymptotic dynamics on the laminar-turbulent boundary can be periodic, or even chaotic. Similar computational studies of shear flows [8][9][10][11][12][13][14] yielded various types of asymptotic states in the laminar-turbulent boundary. In most of the current literature, irrespective of the type of their time-dependence, such asymptotic states at the laminarturbulent boundary are referred to as the "edge states"; and the bisection-based methods that probe edge states are called "edge tracking".…”

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confidence: 81%

Upper edge of chaos and the energetics of transition in pipe flow

et al. 2020

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In the past two decades, our understanding of the transition to turbulence in shear flows with linearly stable laminar solutions has greatly improved. Regarding the susceptibility of the laminar flow, two concepts have been particularly useful: the edge states and the minimal seeds. In this nonlinear picture of the transition, the basin boundary of turbulence is set by the edge state's stable manifold and this manifold comes closest in energy to the laminar equilibrium at the minimal seed. We begin this paper by presenting numerical experiments in which three-dimensional perturbations are too energetic to trigger turbulence in pipe flow but they do lead to turbulence when their amplitude is reduced. We show that this seemingly counter-intuitive observation is in fact consistent with the fully nonlinear description of the transition mediated by the edge state. In order to understand the physical mechanisms behind this process, we measure the turbulent kinetic energy production and dissipation rates as a function of the radial coordinate. Our main observation is that the transition to turbulence relies on the energy amplification away from the wall, as opposed to the turbulence itself, whose energy is predominantly produced near the wall. This observation is further supported by the similar analyses on the minimal seeds and the edge states. Furthermore, we show that the time-evolution of production-over-dissipation curves provide a clear distinction between the different initial amplification stages of the transition to turbulence from the minimal seed. arXiv:1912.09270v1 [physics.flu-dyn] 19 Dec 2019 2

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A spotlike edge state in plane Poiseuille flow

Abstract: We study the laminar turbulent boundary in plane Poiseuille flow at Re = 1400 and 2180 using the technique of edge tracking. For large computational domains the attracting state in the laminar-turbulent boundary is localized in spanwise and streamwise direction and chaotic.

Cited by 5 publications

References 14 publications

Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

Streamwise and doubly-localised periodic orbits in plane Poiseuille flow

State space geometry of the chaotic pilot-wave hydrodynamics

Upper edge of chaos and the energetics of transition in pipe flow

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