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

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Lower branch coherent states in plane Couette flow have an asymptotic structure that consists of O(1) streaks, O(R −1 ) streamwise rolls and a weak sinusoidal wave that develops a critical layer, for large Reynolds number R. Higher harmonics become negligible. These unstable lower branch states appear to have a single unstable eigenvalue at all Reynolds numbers. These results suggest that the lower branch coherent states control transition to turbulence and that they may be promising targets for new turbulence prevention strategies.

We present ten new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new traveling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their 3D-physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space onto suitably chosen 2-or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows. arXiv:0808.3375v2 [physics.flu-dyn]

We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in space. Solutions of different size are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming PDE systems. These new solutions are a step towards extending the dynamical systems view of transitional turbulence to spatially extended flows.The discovery of exact equilibrium and traveling-wave solutions to the full nonlinear Navier-Stokes equations has resulted in much recent progress in understanding the dynamics of linearly stable shear flows such as pipe, channel and plane Couette flow [1,2,3,4]. These exact solutions, together with their entangled stable and unstable manifolds, form a dynamical network that supports chaotic dynamics, so that turbulence can be understood as a walk among unstable solutions [5,6]. Moreover, specific exact solutions are found to be edge states [7], that is, solutions with codimension-1 stable manifolds that locally form the stability boundary between laminar and turbulent dynamics. Thus, exact solutions play a key role both in supporting turbulence and in guiding transition.This emerging dynamical systems viewpoint does not yet capture the full spatio-temporal dynamics of turbulent flows. One major limitation is that exact solutions have mostly been studied in small computational domains with periodic boundary conditions. The small periodic solutions cannot capture the localized structures typically observed in spatially extended flows. For example, pipe flows exhibit localized turbulent puffs. Similarly, in plane Couette flow (PCF), the flow between two parallel walls moving in opposite directions, localized perturbations trigger turbulent spots which then invade the surrounding laminar flow [8,9]. Even more regular long-wavelength spatial patterns such as turbulent stripes have been observed [10]. The known periodic exact solutions cannot capture this rich spatial structure, but they do suggest that localized solutions might be key in understanding the dynamics of spatially extended flows.Spatially localized states are common in a variety of driven dissipative systems. These are often found in a parameter regime of bistability (or at least coexistence) between a spatially uniform state and a spatially periodic pattern, such as occurs in a subcritical pattern forming instability. The localized state then resembles a slug of the pattern embedded in the uniform background. An early explanation of such states is due to Pomeau [11], who argued that a front between a spatially uniform and spatially periodic state, which might otherwise be expected to drift in time, can be stabilized over a finite parameter range by pinning to the spatial phase of the pattern. More recently, the details of this localization mechanism have been established for the subcritical Swift-Hohenberg equation (SHE) through a theory of spatial dynamics [12,13,14]. In one spatial dimension the time-inde...

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We apply the iterated edge-state tracking algorithm to study the boundary between laminar and turbulent dynamics in plane Couette flow at Re= 400. Perturbations that are not strong enough to become fully turbulent or weak enough to relaminarize tend toward a hyperbolic coherent structure in state space, termed the edge state, which seems to be unique up to obvious continuous shift symmetries. The results reported here show that in cases where a fixed point has only one unstable direction, such as for the lower-branch solution in plane Couette flow, the iterated edge tracking algorithm converges to this state. They also show that the choice of initial state is not critical and that essentially arbitrary initial conditions can be used to find the edge state. DOI: 10.1103/PhysRevE.78.037301 PACS number͑s͒: 47.10.Fg, 47.27.Cn, 47.27.ed Plane Couette flow and pipe flow belong to the class of shear flows where turbulence occurs despite the persistent linear stability of the laminar profile ͓1͔. Triggering turbulence then requires the crossing of two thresholds, one in Reynolds number and one in perturbation amplitude. Guidance on the minimum Reynolds number is offered by the appearance of exact coherent states: once they are present, an entanglement of their stable and unstable manifolds can provide the necessary state space elements for chaotic, turbulent dynamics ͓2-14͔. Since the exact coherent states appear in saddle-node bifurcations, it is natural to associate the upper branch ͑characterized by a higher kinetic energy or a higher drag͒ with the turbulent dynamics and the lower branch with the threshold in perturbation amplitude ͓8,15-18͔. In plane Couette flow this scenario seems to be borne out: at the point of bifurcation, at a Reynolds number of about 127.7, the upper branch state is stable and the lower one has only one unstable direction ͓6,8,18͔. At slightly higher Reynolds numbers, the upper branch undergoes secondary bifurcations which could lead to the complex state space structure usually associated with turbulent dynamics. For the lower branch, on the other hand, there are no indications of further bifurcations. If it continues to have a single unstable direction only, its stable manifold can divide the state space such that initial conditions from one side decay more or less directly to the laminar profile, whereas those from the other side show some turbulent dynamics. Such a description of the transition along the lines of the phenomenology of saddle-node bifurcations has been advanced by Toh and Itano ͓15͔ for plane Poiseuille flow, by Wang et al. ͓18͔ and Viswanath ͓19,20͔ for plane Couette flow, and Kerswell and Tutty for pipe flow ͓21͔.Empirically, one may study the boundary between laminar and turbulent dynamics by following the time evolution of flow fields and thereby assigning a lifetime-i.e., the time it takes for a particular initial condition to decay toward the laminar profile ͓22-25͔. Increasing the amplitude of the perturbation one notes changes between regions with smooth variati...

Turbulent-laminar banded patterns in plane Poiseuille flow are studied via direct numerical simulations in a tilted and translating computational domain using a parallel version of the pseudospectral code Channelflow. 3D visualizations via the streamwise vorticity of an instantaneous and a time-averaged pattern are presented, as well as 2D visualizations of the average velocity field and the turbulent kinetic energy. Simulations for 2300 ≥ Re b ≥ 700 show the gradual development from uniform turbulence to a pattern with wavelength 20 half-gaps at Re b ≈ 1900, to a pattern with wavelength 40 at Re b ≈ 1300 and finally to laminar flow at Re b ≈ 800. These transitions are tracked quantitatively via diagnostics using the amplitude and phase of the Fourier transform and its probability distribution. The propagation velocity of the pattern is approximately that of the mean flux and is a decreasing function of Reynolds number. Examination of the time-averaged flow shows that a turbulent band is associated with two counter-rotating cells stacked in the cross-channel direction and that the turbulence is highly concentrated near the walls. Near the wall, the Reynolds stress force accelerates the fluid through a turbulent band while viscosity decelerates it; advection by the laminar profile acts in both directions. In the center, the Reynolds stress force decelerates the fluid through a turbulent band while advection by the laminar profile accelerates it. These characteristics are compared with those of turbulent-laminar banded patterns in plane Couette flow.