129 research outputs found
Shear-flow transition: the basin boundary
The structure of the basin of attraction of a stable equilibrium point is
investigated for a dynamical system (W97) often used to model transition to
turbulence in shear flows. The basin boundary contains not only an equilibrium
point Xlb but also a periodic orbit P, and it is the latter that mediates the
transition. Orbits starting near Xlb relaminarize. We offer evidence that this
is due to the extreme narrowness of the region complementary to basin of
attraction in that part of phase space near Xlb. This leads to a proposal for
interpreting the 'edge of chaos' in terms of more familiar invariant sets.Comment: 11 pages; submitted for publication in Nonlinearit
Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?
The instability of a streak and its nonlinear evolution are investigated by
direct numerical simulation (DNS) for plane Poiseuille flow at Re=3000. It is
suggested that there exists a traveling-wave solution (TWS). The TWS is
localized around one of the two walls and notably resemble to the coherent
structures observed in experiments and DNS so far. The phase space structure
around this TWS is similar to a saddle point. Since the stable manifold of this
TWS is extended close to the quasi two dimensional (Q2D) energy axis, the
approaching process toward the TWS along the stable manifold is approximately
described as the instability of the streak (Q2D flow) and the succeeding
nonlinear evolution. Bursting corresponds to the escape from the TWS along the
unstable manifold. These manifolds constitute part of basin boundary of the
turbulent state.Comment: 5 pages, 6 figure
Visualizing the geometry of state space in plane Couette flow
Motivated by recent experimental and numerical studies of coherent structures
in wall-bounded shear flows, we initiate a systematic exploration of the
hierarchy of unstable invariant solutions of the Navier-Stokes equations. We
construct a dynamical, 10^5-dimensional state-space representation of plane
Couette flow at Re = 400 in a small, periodic cell and offer a new method of
visualizing invariant manifolds embedded in such high dimensions. We compute a
new equilibrium solution of plane Couette flow and the leading eigenvalues and
eigenfunctions of known equilibria at this Reynolds number and cell size. What
emerges from global continuations of their unstable manifolds is a surprisingly
elegant dynamical-systems visualization of moderate-Reynolds turbulence. The
invariant manifolds tessellate the region of state space explored by
transiently turbulent dynamics with a rigid web of continuous and discrete
symmetry-induced heteroclinic connections.Comment: 32 pages, 13 figures submitted to Journal of Fluid Mechanic
Experimental scaling law for the sub-critical transition to turbulence in plane Poiseuille flow
We present an experimental study of transition to turbulence in a plane
Poiseuille flow. Using a well-controlled perturbation, we analyse the flow
using extensive Particule Image Velocimetry and flow visualisation (using Laser
Induced Fluorescence) measurements and use the deformation of the mean velocity
profile as a criterion to characterize the state of the flow. From a large
parametric study, four different states are defined depending on the values of
the Reynolds number and the amplitude of the perturbation. We discuss the role
of coherent structures, like hairpin vortices, in the transition. We find that
the minimal amplitude of the perturbation triggering transition scales like
Re^-1
On a self-sustained process at large scale in the turbulent channel flow
Large-scale motions, important in turbulent shear flows, are frequently
attributed to the interaction of structures at smaller scale. Here we show
that, in a turbulent channel at Re_{\tau} \approx 550, large-scale motions can
self-sustain even when smaller-scale structures populating the near-wall and
logarithmic regions are artificially quenched. This large-scale self-sustained
mechanism is not active in periodic boxes of width smaller than Lz ~ 1.5h or
length shorter than Lx ~ 3h which correspond well to the most energetic large
scales observed in the turbulent channel
Turbulence transition and the edge of chaos in pipe flow
The linear stability of pipe flow implies that only perturbations of
sufficient strength will trigger the transition to turbulence. In order to
determine this threshold in perturbation amplitude we study the \emph{edge of
chaos} which separates perturbations that decay towards the laminar profile and
perturbations that trigger turbulence. Using the lifetime as an indicator and
methods developed in (Skufca et al, Phys. Rev. Lett. {\bf 96}, 174101 (2006))
we show that superimposed on an overall -scaling predicted and studied
previously there are small, non-monotonic variations reflecting folds in the
edge of chaos. By tracing the motion in the edge we find that it is formed by
the stable manifold of a unique flow field that is dominated by a pair of
downstream vortices, asymmetrically placed towards the wall. The flow field
that generates the edge of chaos shows intrinsic chaotic dynamics.Comment: 4 pages, 5 figure
On self-sustaining processes in Rayleigh-stable rotating plane Couette flows and subcritical transition to turbulence in accretion disks
Subcritical transition to turbulence in Keplerian accretion disks is still a
controversial issue and some theoretical progress is required in order to
determine whether or not this scenario provides a plausible explanation for the
origin of angular momentum transport in non-magnetized accretion disks.
Motivated by the recent discoveries of exact nonlinear steady self-sustaining
solutions in linearly stable non-rotating shear flows, we attempt to compute
similar solutions in Rayleigh-stable rotating plane Couette flows and to
identify transition mechanisms in such flows by combining nonlinear
continuation methods and asymptotic theory. We obtain exact nonlinear solutions
for Rayleigh-stable cyclonic regimes but show that it is not possible to
compute solutions for Rayleigh-stable anticyclonic regimes, including Keplerian
flow, using similar techniques. We also present asymptotic descriptions of
these various problems at large Reynolds numbers that provide some insight into
the differences between the non-rotating and Rayleigh-stable anticyclonic
regimes and derive some necessary conditions for mechanisms analogous to the
non-rotating self-sustaining process to be present in flows on the Rayleigh
line. Our results demonstrate that subcritical transition mechanisms cannot be
identified in wall-bounded Rayleigh-stable anticyclonic shear flows by
transposing directly the phenomenology of subcritical transition in cyclonic
and non-rotating wall-bounded shear flows. Asymptotic developments, however,
leave open the possibility that nonlinear self-sustaining solutions may exist
in unbounded or periodic flows on the Rayleigh line. These could serve as a
starting point to discover solutions in Rayleigh-stable flows, but the
nonlinear stability of Keplerian accretion disks remains to be determined.Comment: 16 pages, 12 figures. Accepted for publication in A&
A Streamwise Constant Model of Turbulence in Plane Couette Flow
Streamwise and quasi-streamwise elongated structures have been shown to play
a significant role in turbulent shear flows. We model the mean behavior of
fully turbulent plane Couette flow using a streamwise constant projection of
the Navier Stokes equations. This results in a two-dimensional, three velocity
component () model. We first use a steady state version of the model to
demonstrate that its nonlinear coupling provides the mathematical mechanism
that shapes the turbulent velocity profile. Simulations of the model
under small amplitude Gaussian forcing of the cross-stream components are
compared to DNS data. The results indicate that a streamwise constant
projection of the Navier Stokes equations captures salient features of fully
turbulent plane Couette flow at low Reynolds numbers. A system theoretic
approach is used to demonstrate the presence of large input-output
amplification through the forced model. It is this amplification
coupled with the appropriate nonlinearity that enables the model to
generate turbulent behaviour under the small amplitude forcing employed in this
study.Comment: Journal of Fluid Mechanics 2010, in pres
Pattern fluctuations in transitional plane Couette flow
In wide enough systems, plane Couette flow, the flow established between two
parallel plates translating in opposite directions, displays alternatively
turbulent and laminar oblique bands in a given range of Reynolds numbers R. We
show that in periodic domains that contain a few bands, for given values of R
and size, the orientation and the wavelength of this pattern can fluctuate in
time. A procedure is defined to detect well-oriented episodes and to determine
the statistics of their lifetimes. The latter turn out to be distributed
according to exponentially decreasing laws. This statistics is interpreted in
terms of an activated process described by a Langevin equation whose
deterministic part is a standard Landau model for two interacting complex
amplitudes whereas the noise arises from the turbulent background.Comment: 13 pages, 11 figures. Accepted for publication in Journal of
statistical physic
Large scale flow effects, energy transfer, and self-similarity on turbulence
The effect of large scales on the statistics and dynamics of turbulent
fluctuations is studied using data from high resolution direct numerical
simulations. Three different kinds of forcing, and spatial resolutions ranging
from 256^3 to 1024^3, are being used. The study is carried out by investigating
the nonlinear triadic interactions in Fourier space, transfer functions,
structure functions, and probability density functions. Our results show that
the large scale flow plays an important role in the development and the
statistical properties of the small scale turbulence. The role of helicity is
also investigated. We discuss the link between these findings and
intermittency, deviations from universality, and possible origins of the
bottleneck effect. Finally, we briefly describe the consequences of our results
for the subgrid modeling of turbulent flows
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