42 research outputs found
Spatiotemporal dynamics in 2D Kolmogorov flow over large domains
Kolmogorov flow in two dimensions - the two-dimensional Navier-Stokes
equations with a sinusoidal body force - is considered over extended periodic
domains to reveal localised spatiotemporal complexity. The flow response
mimicks the forcing at small forcing amplitudes but beyond a critical value
develops a long wavelength instability. The ensuing state is described by a
Cahn-Hilliard-type equation and as a result coarsening dynamics are observed
for random initial data. After further bifurcations, this regime gives way to
multiple attractors, some of which possess spatially-localised time dependence.
Co-existence of such attractors in a large domain gives rise to interesting
collisional dynamics which is captured by a system of 5 (1-space and 1-time)
PDEs based on a long wavelength limit. The coarsening regime reinstates itself
at yet higher forcing amplitudes in the sense that only longest-wavelength
solutions remain attractors. Eventually, there is one global longest-wavelength
attractor which possesses two localised chaotic regions - a kink and antikink -
which connect two steady one-dimensional flow regions of essentially half the
domain width each. The wealth of spatiotemporal complexity uncovered presents a
bountiful arena in which to study the existence of simple invariant localised
solutions which presumably underpin all of the observed behaviour
Reply to Comment on 'Critical behaviour in the relaminarization of localized turbulence in pipe flow'
This is a Reply to Comment arXiv:0707.2642 by Hof et al. on Letter
arXiv:physics/0608292 which was subsequently published in Phys Rev Lett, 98,
014501 (2007).
In our letter it was reported that in pipe flow the median time for
relaminarisation of localised turbulent disturbances closely follows the
scaling . This conclusion was based on data from
collections of 40 to 60 independent simulations at each of six different
Reynolds numbers, Re. In the Comment, Hof et al. estimate differently
for the point at lowest Re. Although this point is the most uncertain, it forms
the basis for their assertion that the data might then fit an exponential
scaling , for some constant A, supporting Hof et al.
(2006) Nature, 443, 59. The most certain point (at largest Re) does not fit
their conclusion and is rejected. We clarify why their argument for rejecting
this point is flawed. The median is estimated from the distribution of
observations, and it is shown that the correct part of the distribution is
used. The data is sufficiently well determined to show that the exponential
scaling cannot be fit to the data over this range of Re, whereas the fit is excellent, indicating critical behaviour and supporting
experiments by Peixinho & Mullin 2006.Comment: 1 page, 1 figur
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Exhausting the background approach for bounding the heat transport in Rayleigh-Bénard convection
We revisit the optimal heat transport problem for Rayleigh-B\'enard convection in which a rigorous upper bound on the Nusselt number, , is sought as a function of the Rayleigh number . Concentrating on the 2-dimensional problem with stress-free boundary conditions, we impose the full heat equation as a constraint for the bound using a novel 2-dimensional background approach thereby complementing the `wall-to-wall' approach of Hassanzadeh \etal \,(\emph{J. Fluid Mech.} \textbf{751}, 627-662, 2014). Imposing the same symmetry on the problem, we find correspondence with their result for but, beyond that, the optimal fields complexify to produce a higher bound. This bound approaches that by a 1-dimensional background field as the length of computational domain . On lifting the imposed symmetry, the optimal 2-dimensional temperature background field reverts back to being 1-dimensional giving the best bound compared to in the non-slip case. % We then show via an inductive bifurcation analysis that imposing the full time-averaged Boussinesq equations as constraints (by introducing 2-dimensional temperature {\em and} velocity background fields) is also unable to lower this bound. This then exhausts the background approach for the 2-dimensional (and by extension 3-dimensional) Rayleigh-Benard problem with the bound remaining stubbornly while data seems more to scale like for large . % Finally, we show that adding a velocity background field to the formulation of Wen \etal\, (\emph{Phys. Rev. E.} \textbf{92}, 043012, 2015), which is able to use an extra vorticity constraint due to the stress-free condition to lower the bound to , also fails to improve the bound.EPSRC under grant EP/P001130/1
Connection between nonlinear energy optimization and instantons.
How systems transit between different stable states under external perturbation is an important practical issue. We discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems. In the context of the one-dimensional Swift-Hohenberg equation, which has multiple stable equilibria, we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances-minimal seeds-for transition to specific attractors from the ground state. Then, after generalizing the technique to consider multiple, equally spaced-in-time perturbations, it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations. Importantly, we find that the key features of the instanton can be captured by a low number of discrete perturbations (typically one perturbation per basin of attraction crossed). This suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton
Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow
Motivated by recent success in the dynamical systems approach to transitional
flow, we study the efficiency and effectiveness of extracting simple invariant
sets (recurrent flows) directly from chaotic/turbulent flows and the potential
of these sets for providing predictions of certain statistics of the flow.
Two-dimensional Kolmogorov flow (the 2D Navier-Stokes equations with a
sinusoidal body force) is studied both over a square [0, 2{\pi}]2 torus and a
rectangular torus extended in the forcing direction. In the former case, an
order of magnitude more recurrent flows are found than previously (Chandler &
Kerswell 2013) and shown to give improved predictions for the dissipation and
energy pdfs of the chaos via periodic orbit theory. Over the extended torus at
low forcing amplitudes, some extracted states mimick the statistics of the
spatially-localised chaos present surprisingly well recalling the striking
finding of Kawahara & Kida (2001) in low-Reynolds-number plane Couette flow. At
higher forcing amplitudes, however, success is limited highlighting the
increased dimensionality of the chaos and the need for larger data sets.
Algorithmic developments to improve the extraction procedure are discussed