163 research outputs found
Understanding the sub-critical transition to turbulence in wall flows
Contrasting with free shear flows presenting velocity profiles with
inflection points which cascade to turbulence in a relatively mild way, wall
bounded flows are deprived of (inertial) instability modes at low Reynolds
numbers and become turbulent in a much wilder way, most often marked by the
coexistence of laminar and turbulent domains at intermediate Reynolds numbers,
well below the range where (viscous) instabilities can show up. There can even
be no unstable mode at all, as for plane Couette flow (pCf) or for Poiseuille
pipe flow (Ppf) that currently are the subject of intense research. Though the
mechanisms involved in the transition to turbulence in wall flows are now
better understood, statistical properties of the transition itself are yet
unsatisfactorily assessed. A review of the situation is given. An alternative
to the temporal theory of the transition to turbulence in terms of chaotic
transients in such globally subcritical flows is proposed, which invokes
spatio-temporal intermittence and the theory of first order (thermodynamic)
phase transitions.Comment: 15 pages, 7 figures; proceedings of the Conference and Workshop on
Perspectives in Nonlinear Dynamics, Trieste, 200
On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular
The main part of this contribution to the special issue of EJM-B/Fluids
dedicated to Patrick Huerre outlines the problem of the subcritical transition
to turbulence in wall-bounded flows in its historical perspective with emphasis
on plane Couette flow, the flow generated between counter-translating parallel
planes. Subcritical here means discontinuous and direct, with strong
hysteresis. This is due to the existence of nontrivial flow regimes between the
global stability threshold Re_g, the upper bound for unconditional return to
the base flow, and the linear instability threshold Re_c characterized by
unconditional departure from the base flow. The transitional range around Re_g
is first discussed from an empirical viewpoint ({\S}1). The recent
determination of Re_g for pipe flow by Avila et al. (2011) is recalled. Plane
Couette flow is next examined. In laboratory conditions, its transitional range
displays an oblique pattern made of alternately laminar and turbulent bands, up
to a third threshold Re_t beyond which turbulence is uniform. Our current
theoretical understanding of the problem is next reviewed ({\S}2): linear
theory and non-normal amplification of perturbations; nonlinear approaches and
dynamical systems, basin boundaries and chaotic transients in minimal flow
units; spatiotemporal chaos in extended systems and the use of concepts from
statistical physics, spatiotemporal intermittency and directed percolation,
large deviations and extreme values. Two appendices present some recent
personal results obtained in plane Couette flow about patterning from numerical
simulations and modeling attempts.Comment: 35 pages, 7 figures, to appear in Eur. J. Mech B/Fluid
Turbulent patterns in wall-bounded flows: a Turing instability?
In their way to/from turbulence, plane wall-bounded flows display an
interesting transitional regime where laminar and turbulent oblique bands
alternate, the origin of which is still mysterious. In line with Barkley's
recent work about the pipe flow transition involving reaction-diffusion
concepts, we consider plane Couette flow in the same perspective and transform
Waleffe's classical four-variable model of self-sustaining process into a
reaction-diffusion model. We show that, upon fulfillment of a condition on the
relative diffusivities of its variables, the featureless turbulent regime
becomes unstable against patterning as the result of a Turing instability. A
reduced two-variable model helps us to delineate the appropriate region of
parameter space. An {\it intrinsic} status is therefore given to the pattern's
wavelength for the first time. Virtues and limitations of the model are
discussed, calling for a microscopic support of the phenomenological approach.Comment: to appear in Europhysics Letters in a different forma
Spatiotemporal perspective on the decay of turbulence in wall-bounded flows
Using a reduced model focusing on the in-plane dependence of plane Couette
flow, it is shown that the turbulent-to-laminar relaxation process can be
understood as a nucleation problem similar to that occurring at a thermodynamic
first-order phase transition. The approach, apt to deal with the large
extension of the system considered, challenges the current interpretation in
terms of chaotic transients typical of temporal chaos. The study of the
distribution of the sizes of laminar domains embedded in turbulent flow proves
that an abrupt transition from sustained spatiotemporal chaos to laminar flow
can take place at some given value of the Reynolds number R_{low}, whether or
not the local chaos lifetime, as envisioned within low-dimensional dynamical
systems theory, diverges at finite R beyond R_{low}.Comment: 9 pages, 3 figures, published in 2009 as a Rapid Communication in
Phys. Rev. E, vol. 79, article 025301, corrected to include erratum Phys.
Rev. E 79, 039904. References to now published material have been updated. A
note has been added pointing to recent related work by D. Barkley
(arXiv:1101.4125v1
On modelling transitional turbulent flows using under-resolved direct numerical simulations: The case of plane Couette flow
Direct numerical simulations have proven of inestimable help to our
understanding of the transition to turbulence in wall-bounded flows. While the
dynamics of the transition from laminar flow to turbulence via localised spots
can be investigated with reasonable computing resources in domains of limited
extent, the study of the decay of turbulence in conditions approaching those in
the laboratory requires consideration of domains so wide as to exclude the
recourse to fully resolved simulations. Using Gibson's C++ code ChannelFlow, we
scrutinize the effects of a controlled lowering of the numerical resolution on
the decay of turbulence in plane Couette flow at a quantitative level. We show
that the number of Chebyshev polynomials describing the cross-stream dependence
can be drastically decreased while preserving all the qualitative features of
the solution. In particular, the oblique turbulent band regime experimentally
observed in the upper part of the transitional range is extremely robust. In
terms of Reynolds numbers, the resolution lowering is seen to yield a regular
downward shift of the upper and lower thresholds Rt and Rg where the bands
appear and break down. The study is illustrated with the results of two
preliminary experiments.Comment: 20 pages, 9 figures. Accepted on August 24, 2010, to appear in TCF
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