Dynamics of spatially localized states in transitional plane Couette flow

Pershin, Anton; Beaume, Cédric; Tobias, Steven M.

doi:10.1017/jfm.2019.154

Cited by 7 publications

(6 citation statements)

References 48 publications

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“…From an algorithmic view, developing the far-field core decomposition idea to be used with timestepping bifurcation routines would allow for the investigation of various invasion front instabilities found in doubly diffusive convection and plane Couette flow [5,48]. We note that the almost planar depinning fronts are similar to the twist instability seen by [5] close to the snaking region and where stability is regained as one moves sufficiently far from the snaking region.…”

Section: Discussionmentioning

confidence: 87%

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

Lloyd¹

2019

Preprint

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It is well-known that stationary localised patterns involving a periodic stripe core can undergo a process that is known as 'homoclinic snaking' where patterns are added to the stripe core as a bifurcation parameter is varied. The parameter region where homoclinic snaking takes place usually occupies a small region in the bistability region between the stripes and quiescent state. Outside the homoclinic snaking region, the localised patterns invade or retreat where stripes are either added or removed from the core forming depinning fronts. It remains an open problem to carry out a numerical bifurcation analysis of depinning fronts. In this paper, we carry out a numerical bifurcation analysis of depinning of fronts near the homoclinic snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in the two-dimensional Swift-Hohenberg equation with either a quadratic-cubic or cubic-quintic nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel, oblique and perpendicular to the front interface, and almost planar depinning fronts. We show that invading parallel depinning fronts select both a far-field wavenumber and a propagation wavespeed whereas retreating parallel depinning fronts come in families where the wavespeed is a function of the far-field wavenumber. Employing a far-field core decomposition, we propose a boundary value problem for the invading depinning fronts which we numerically solve and use path-following routines to trace out bifurcation diagrams. We then carry out a thorough numerical investigation of the parallel, oblique, perpendicular stripe, and almost planar invasion fronts. We find that almost planar invasion fronts in the cubic-quintic Swift-Hohenberg equation bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region. Sufficiently far from the 1D homoclinic snaking region, no almost planar invasion fronts exist and we find that parallel invasion stripe fronts may regain transverse stability if they propagate above a critical speed. Finally, we show that depinning fronts shed light on the time simulations of fully localised patches of stripes on the plane. The numerical algorithms detailed have wider application to general modulated fronts and reaction-diffusion systems.

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Section: Discussionmentioning

confidence: 87%

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

Lloyd¹

2019

Preprint

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“…the flow confined between two infinite plates separated by a gap 2h and moving in opposite directions with constant velocity U. The domain is periodic in the streamwise (period Γ x = 4πh) and in the spanwise (period Γ z = 32πh/15) directions, and no-slip boundary conditions are applied at the walls (Pershin, Beaume & Tobias 2019). The non-dimensionalized Navier-Stokes equation and the incompressibility condition are given by…”

Section: Protocolmentioning

confidence: 99%

“…on the numerical resolution used in Pershin et al (2019). The temporal discretization is performed by means of third-order semi-implicit backward differentiation with time step t = 1/Re.…”

Section: Protocolmentioning

confidence: 99%

A probabilistic protocol for the assessment of transition and control

2020

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“…Riols et ala [37], in a study of subcritical transition to magnetorotational dynamo in Keplerian shear flows, computed both homoclinic and heteroclinic connections between unstable POs. Pershin et ala [38] found a heteroclinic connection from an EQ to a nearby PO in PCF. In both Riols et alaand Pershin et al, connections were computed very close to the saddle-node bifurcations leading to the formation of these invariant solutions.…”

Section: Introductionmentioning

confidence: 96%

Heteroclinic and homoclinic connections in a Kolmogorov-like flow

et al. 2019

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Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions -equilibria, periodic, and quasi-periodic orbits -as well as continua of connections forming higher-dimensional connecting manifolds. We also compute a homoclinic connection of a periodic orbit and provide strong evidence that the associated homoclinic tangle forms the chaotic repeller that underpins transient turbulence in the symmetric subspace.arXiv:1907.05860v1 [physics.flu-dyn]

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Dynamics of spatially localized states in transitional plane Couette flow

Cited by 7 publications

References 48 publications

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

Invasion Fronts Outside the Homoclinic Snaking Region in the Planar Swift-Hohenberg Equation

A probabilistic protocol for the assessment of transition and control

Heteroclinic and homoclinic connections in a Kolmogorov-like flow

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