59 research outputs found
Low-dimensional models for turbulent plane Couette flow in a minimal flow unit
We model turbulent plane Couette flow in the minimal flow unit (MFU) – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence – by expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier–Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on ‘uncoupled’ eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier–Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the ‘backbone’ for MFU turbulence is a periodic orbit, and support the roll–streak–breakdown–roll reformation picture of shear-driven turbulence
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
Statistical Properties of Turbulence: An Overview
We present an introductory overview of several challenging problems in the
statistical characterisation of turbulence. We provide examples from fluid
turbulence in three and two dimensions, from the turbulent advection of passive
scalars, turbulence in the one-dimensional Burgers equation, and fluid
turbulence in the presence of polymer additives.Comment: 34 pages, 31 figure
Basin boundary, edge of chaos, and edge state in a two-dimensional model
In shear flows like pipe flow and plane Couette flow there is an extended
range of parameters where linearly stable laminar flow coexists with a
transient turbulent dynamics. When increasing the amplitude of a perturbation
on top of the laminar flow, one notes a a qualitative change in its lifetime,
from smoothly varying and short one on the laminar side to sensitively
dependent on initial conditions and long on the turbulent side. The point of
transition defines a point on the edge of chaos. Since it is defined via the
lifetimes, the edge of chaos can also be used in situations when the turbulence
is not persistent. It then generalises the concept of basin boundaries, which
separate two coexisting attractors, to cases where the dynamics on one side
shows transient chaos and almost all trajectories eventually end up on the
other side. In this paper we analyse a two-dimensional map which captures many
of the features identified in laboratory experiments and direct numerical
simulations of hydrodynamic flows. The analysis of the map shows that different
dynamical situations in the edge of chaos can be combined with different
dynamical situations in the turbulent region. Consequently, the model can be
used to develop and test further characterisations that are also applicable to
realistic flows.Comment: 24 pages, 9 color figure
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