arXiv subjects

D. Barkley

Publications and source records attributed to D. Barkley.

9 recordsLinked to original sources

arXiv2022

Patterns in transitional shear turbulence. Part 2: Emergence and optimal wavelength

Low Reynolds number turbulence in wall-bounded shear flows \emph{en route} to laminar flow takes the form of oblique, spatially-intermittent turbulent structures. In plane Couette flow, these emerge from uniform turbulence via a spatiotemporal intermittent process in which localised quasi-laminar gaps randomly nucleate and disappear. For slightly lower Reynolds numbers, spatially periodic and approximately stationary turbulent-laminar patterns predominate. The statistics of quasi-laminar regions, including the distributions of space and time scales and their Reynolds number dependence, are analysed. A smooth, but marked transition is observed between uniform turbulence and flow with intermittent quasi-laminar gaps, whereas the transition from gaps to regular patterns is more gradual. Wavelength selection in these patterns is analysed via numerical simulations in oblique domains of various sizes. Via lifetime measurements in minimal domains, and a wavelet-based analysis of wavelength predominance in a large domain, we quantify the existence and non-linear stability of a pattern as a function of wavelength and Reynolds number. We report that the preferred wavelength maximises the energy and dissipation of the large-scale flow along laminar-turbulent interfaces. This optimal behaviour is primarily due to the advective nature of the large-scale flow, with turbulent fluctuations playing only a secondary role.

physics.flu-dyn↗

arXiv2022

Patterns in transitional shear turbulence. Part 1. Energy transfer and mean-flow interaction

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of spatially intermittent turbulent structures. In plane shear flows, these appear as a regular pattern of alternating turbulent and quasi-laminar flow. Both the physical and the spectral energy balance of a turbulent-laminar pattern in plane Couette flow are computed and compared to those of uniform turbulence. In the patterned state, the mean flow is strongly modulated and is fuelled by two mechanisms: primarily, the nonlinear self-interaction of the mean flow (via mean advection), and secondly, the extraction of energy from turbulent fluctuations (via negative spectral production, associated with an energy transfer from small to large scales). Negative production at large scales is also found in the uniformly turbulent state. Important features of the energy budgets are surveyed as a function of $Re$ through the transition between uniform turbulence and turbulent-laminar patterns.

physics.flu-dyn↗

arXiv2022

Extreme events in transitional turbulence

Transitional localised turbulence in shear flows is known to either decay to an absorbing laminar state or proliferate via splitting. The average passage times from one state to the other depend super-exponentially on the Reynolds number and lead to a crossing Reynolds number above which proliferation is more likely than decay. In this paper, we apply a rare event algorithm, Adaptative Multilevel Splitting (AMS), to the deterministic Navier-Stokes equations to study transition paths and estimate large passage times in channel flow more efficiently than direct simulations. We establish a connection with extreme value distributions and show that transition between states is mediated by a regime that is self-similar with the Reynolds number. The super-exponential variation of the passage times is linked to the Reynolds-number dependence of the parameters of the extreme value distribution. Finally, motivated by instantons from Large Deviation Theory, we show that decay or splitting events approach a most-probable pathway.

physics.flu-dyn↗

arXiv2010

Influence of counter-rotating von Karman flow on cylindrical Rayleigh-Benard convection

The axisymmetric flow in an aspect-ratio-one cylinder whose upper and lower bounding disks are maintained at different temperatures and rotate at equal and opposite velocities is investigated. In this combined Rayleigh-Benard/von Karman problem, the imposed temperature gradient is measured by the Rayleigh number Ra and the angular velocity by the Reynolds number Re. Although fluid motion is present as soon as Re is non-zero, a symmetry-breaking transition analogous to the onset of convection takes place at a finite Rayleigh number higher than that for Re=0. For Re<95, the transition is a pitchfork bifurcation to a pair of steady states, while for Re>95, it is a Hopf bifurcation to a limit cycle. The steady states and limit cycle are connected via a pair of SNIPER bifurcations except very near the Takens-Bogdanov codimension-two point, where the scenario includes global bifurcations. Detailed phase portraits and bifurcation diagrams are presented, as well as the evolution of the leading part of the spectrum, over the parameter ranges 0<Re<120 and 0<Ra<30,000.

physics.flu-dyn↗

arXiv2010

Computation of the Drift Velocity of Spiral Waves using Response Functions

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. In the presence of a small perturbation, the spiral wave's centre of rotation and fiducial phase may change over time, i.e. the spiral wave drifts. In linear approximation, the velocity of the drift is proportional to the convolution of the perturbation with the spiral's Response Functions (RFs), which are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues $λ= 0, \pm iω$. Here we demonstrate that the response functions give quantitatively accurate prediction of the drift velocities due to a variety of perturbations: a time dependent, periodic perturbation (inducing resonant drift); a rotational symmetry breaking perturbation (inducing electrophoretic drift); and a translational symmetry breaking perturbation (inhomogeneity induced drift) including drift due to a gradient, step-wise and localised inhomogeneity. We predict the drift velocities using the response functions in FitzHugh-Nagumo (FHN) and Barkley models, and compare them with the velocities obtained in direct numerical simulations. In all cases good quantitative agreement is demonstrated.

nlin.PS↗

arXiv2010

Orbital movement of spiral waves

Spiral waves in active media react to small perturbations as particle-like objects. Here we apply the asymptotic theory to the interaction of spiral waves with a localized inhomogeneity, which leads to a novel prediction: drift of the spiral rotation centre along circular orbits around the inhomogeneity. The stationary orbits have alternating stability and fixed radii, determined by the properties of the bulk medium and the type of inhomogeneity, while the drift speed along an orbit depends on the strength of the inhomogeneity. Direct simulations confirm the validity and robustness of the theoretical predictions and show that these unexpected effects should be observable in experiment.

nlin.PS↗

arXiv2009

Computation of the response functions of spiral waves in active media

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. A small perturbation causes gradual change in spatial location of spiral's rotation center and frequency, i.e. drift. The response functions (RFs) of a spiral wave are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues $λ= 0, \pm iω$. The RFs describe the spiral's sensitivity to small perturbations in the way that a spiral is insensitive to small perturbations where its RFs are close to zero. The velocity of a spiral's drift is proportional to the convolution of RFs with the perturbation. Here we develop a regular and generic method of computing the RFs of stationary rotating spirals in reaction-diffusion equations. We demonstrate the method on the FitzHugh-Nagumo system and also show convergence of the method with respect to the computational parameters, i.e. discretization steps and size of the medium. The obtained RFs are localized at the spiral's core.

nlin.PS↗

arXiv2005

The Moment Map: Nonlinear dynamics of density evolution via a few moments

We explore situations in which certain stochastic and high-dimensional deterministic systems behave effectively as low-dimensional dynamical systems. We define and study moment maps, maps on spaces of low-order moments of evolving distributions, as a means of understanding equations-free multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath.

cond-mat.other↗

arXiv2005

Computation of Spiral Spectra

A computational linear stability analysis of spiral waves in a reaction-diffusion equation is performed on large disks. As the disk radius R increases, eigenvalue spectra converge to the absolute spectrum predicted by Sandstede and Scheel. The convergence rate is consistent with 1/R, except possibly near the edge of the spectrum. Eigenfunctions computed on large disks are compared with predicted exponential forms. Away from the edge of the absolute spectrum the agreement is excellent, while near the edge computed eigenfunctions deviate from predictions, probably due to finite-size effects. In addition to eigenvalues associated with the absolute spectrum, computations reveal point eigenvalues. The point eigenvalues and associated eigenfunctions responsible for both core and far-field breakup of spiral waves are shown.

nlin.PS↗