Streamwise-localized solutions at the onset of turbulence in pipe flow

Following marginal stability manifolds in quasilinear dynamical reductions of multiscale flows in two space dimensions

We derive a two-dimensional (2D) extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities. The emergent dynamics of these systems is characterized by a slow evolution of (suitably defined) mean fields coupled to marginally stable, fast fluctuation fields. By exploiting this scale separation, an efficient hybrid fast-eigenvalue/slow-initial-value solution algorithm can be developed in which the amplitude of the fast fluctuations is slaved to the slowly evolving mean fields to ensure marginal stability-and temporal scale separation-is maintained. For 2D systems, the fluctuation eigenfunctions are labeled by their Fourier wave numbers characterizing spatial variability in that extended spatial direction, and the marginal mode(s) must coincide with the fastest-growing mode(s) over all admissible Fourier wave numbers. Here we derive an ordinary differential equation governing the slow evolution of the wave number of the fastest-growing fluctuation mode that simultaneously must be slaved to the mean dynamics to ensure the mode has zero growth rate. We illustrate the procedure in the context of a 2D model partial differential equation that shares certain attributes with the equations governing strongly stratified shear flows and other strongly constrained forms of geophysical turbulence in extreme parameter regimes. The slaved evolution follows one or more marginal stability manifolds, which constitute select state-space structures that are not invariant under the full flow dynamics yet capture quasicoherent structures in physical space in a manner analogous to invariant solutions identified in, e.g., transitionally turbulent shear flows. Accordingly, we propose that marginal stability manifolds are central organizing structures in a dynamical systems description of certain classes of multiscale flows in which scale separation justifies a QL approximation of the dynamics.

Predictive Framework for Flow Reversals and Excursions in Turbulence

We present a dynamical framework for intermittent reversals and excursions (R&Es) of large-scale circulations in turbulence. We show that R&Es can occur when turbulent trajectories in phase space shadow invariant manifolds of certain unstable periodic orbits (UPOs). Consequently, substantial flow reorganization and extreme fluctuations in flow metrics observed during R&Es can be reconstructed by splicing the unstable manifolds of such dynamically relevant UPOs. Using this geometrical framework, we predict imminent R&Es and preemptively avert these extreme events using closed-loop control.

Direct Path from Turbulence to Time-Periodic Solutions

Viscous flows through pipes and channels are steady and ordered until, with increasing velocity, the laminar motion catastrophically breaks down and gives way to turbulence. How this apparently discontinuous change from low- to high-dimensional motion can be rationalized within the framework of the Navier-Stokes equations is not well understood. Exploiting geometrical properties of transitional channel flow we trace turbulence to far lower Reynolds numbers (Re) than previously possible and identify the complete path that reversibly links fully turbulent motion to an invariant solution. This precursor of turbulence destabilizes rapidly with Re, and the accompanying explosive increase in attractor dimension effectively marks the transition between deterministic and de facto stochastic dynamics.

Homoclinic bifurcation and switching of edge state in plane Couette flow

We identify the presence of three homoclinic bifurcations that are associated with edge states in a system that is governed by the full Navier-Stokes equation. In plane Couette flow with a streamwise period slightly longer than the minimal unit, we describe a rich bifurcation scenario that is related to new time-periodic solutions and the Nagata steady solution [M. Nagata, J. Fluid Mech. 217, 519-527 (1990)]. In this computational domain, the vigorous time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence and the lower branch of the Nagata steady solution are considered as edge states at different ranges of Reynolds number. These edge states can help in understanding the mechanism of subcritical transition to turbulence in wall-bounded shear flows. At the Reynolds numbers at which the homoclinic bifurcations occur, we find the creation (or destruction) of the time-periodic solutions. At a higher Reynolds number, we observe the edge state switching from the lower-branch Nagata steady solution to PO3 at the creation of this vigorous cycle due to the homoclinic bifurcation. Consequently, the formation of the boundary separating the basins of attraction of the laminar attractor and the time-periodic/chaotic attractor also switches to the respective stable manifolds of the edge states, providing a change in the behavior of a typical amplitude of perturbation toward triggering the transition to turbulence.

Mechanism for turbulence proliferation in subcritical flows

The subcritical transition to turbulence, as occurs in pipe flow, is believed to generically be a phase transition in the directed percolation universality class. At its heart is a balance between the decay rate and proliferation rate of localized turbulent structures, called puffs in pipe flow. Here, we propose the first-ever dynamical mechanism for puff proliferation—the process by which a puff splits into two. In the first stage of our mechanism, a puff expands into a slug. In the second stage, a laminar gap is formed within the turbulent core. The notion of a split-edge state, mediating the transition from a single puff to a two-puff state, is introduced and its form is predicted. The role of fluctuations in the two stages of the transition, and how splits could be suppressed with increasing Reynolds number, are discussed. Using numerical simulations, the mechanism is validated within the stochastic Barkley model. Concrete predictions to test the proposed mechanism in pipe and other wall-bounded flows, and implications for the universality of the directed percolation picture, are discussed.

Dynamical landscape of transitional pipe flow

The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge-states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge. We also discuss alternatives to the suggested bifurcation diagram, which could be relevant for wall-bounded flows other than straight pipes.

Laminar-Turbulent Intermittency in Annular Couette-Poiseuille Flow: Whether a Puff Splits or Not

Direct numerical simulations were carried out with an emphasis on the intermittency and localized turbulence structure occurring within the subcritical transitional regime of a concentric annular Couette-Poiseuille flow. In the annular system, the ratio of the inner to outer cylinder radius is an important geometrical parameter affecting the large-scale nature of the intermittency. We chose a low radius ratio of 0.1 and imposed a constant pressure gradient providing practically zero shear on the inner cylinder such that the base flow was approximated to that of a circular pipe flow. Localized turbulent puffs, that is, axial uni-directional intermittencies similar to those observed in the transitional circular pipe flow, were observed in the annular Couette-Poiseuille flow. Puff splitting events were clearly observed rather far from the global critical Reynolds number, near which given puffs survived without a splitting event throughout the observation period, which was as long as 104 outer time units. The characterization as a directed-percolation universal class was also discussed.

Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.

Basis for finding exact coherent states

One of the outstanding problems in the dynamical systems approach to turbulence is to find a sufficient number of invariant solutions to characterize the underlying dynamics of turbulence [Annu. Rev. Fluid Mech. 44, 203 (2012)10.1146/annurev-fluid-120710-101228]. As a practical matter, the solutions can be difficult to find. To improve this situation, we show how to find periodic orbits and equilibria in plane Couette flow by projecting pseudorecurrent segments of turbulent trajectories onto the left-singular vectors of the Navier-Stokes equations linearized about the relevant mean flow (resolvent modes). The projections are, subsequently, used to initiate Newton-Krylov-hookstep searches, and new (relative) periodic orbits and equilibria are discovered. We call the process project-then-search and validate the process by first applying it to previously known fixed point and periodic solutions. Along the way, we find new branches of equilibria, which include bifurcations from previously known branches, and new periodic orbits that closely shadow turbulent trajectories in state space.

Origin of localized snakes-and-ladders solutions of plane Couette flow

Spatially localized invariant solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations [Schneider, Gibson, and Burke, Phys. Rev. Lett. 104, 104501 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.104501]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of wavy-vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a close connection between pattern-formation theory and Navier-Stokes flow.

Heteroclinic and homoclinic connections in a Kolmogorov-like flow

Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions-equilibria, periodic, and quasiperiodic orbits-as well as continua of connections forming higher-dimensional connecting manifolds. We also compute a homoclinic connection of a periodic orbit and provide strong evidence that the associated homoclinic tangle forms the chaotic repeller that underpins transient turbulence in the symmetric subspace.

Intermittent direction reversals of moving spatially localized turbulence observed in two-dimensional Kolmogorov flow

We have found that in two-dimensional Kolmogorov flow a spatially localized turbulent state (SLT) exists stably and travels with a constant speed on average switching the moving direction randomly and intermittently for moderate values of the control parameters: Reynolds number and the flow rate. We define the coarse-grained position and velocity of an SLT and separate the motion of the SLT from its internal turbulent dynamics by introducing a co-moving frame. The switching process of an SLT represented by the coarse-grained velocity seems to be a random telegraph signal. Focusing on the asymmetry of the internal turbulence we introduce two coarse-grained variables characterizing the internal dynamics. These quantities follow the switching process reasonably. This suggests that the twin attracting invariant sets each of which corresponds to a one-way traveling SLT are embedded in the attractor of the moving SLT and the connection of the two sets is too complicated to be represented by a few degrees of freedom but the motion of an SLT is correlated with the internal turbulent dynamics.

Streamwise decay of localized states in channel flow

Channel flow, the pressure driven flow between parallel plates, has exact coherent structures that show various degrees of localization. For states which are localized in streamwise direction but extended in spanwise direction, we show that they are exponentially localized, with decay constants that are different on the upstream and downstream sides. We extend the analysis of Brand and Gibson [J. Fluid Mech. 750, R1 (2014)]JFLSA70022-112010.1017/jfm.2014.285 for stationary states to the case of advected structures that is needed here, and derive expressions for the decay in terms of eigenvalues and eigenfunctions of certain second order differential equations. The results are in very good agreement with observations on exact coherent structures of different transversal wavelengths.

Chaotic self-sustaining structure embedded in the turbulent-laminar interface

An interface structure between turbulence and laminar flow is investigated in two-dimensional channel flow. This spatially localized structure not only sustains itself but also converts the laminar state into turbulence actively. A filtered simulation technique is introduced to understand the invading process as an inhomogeneity-induced self-sustaining coherent structure, which consists of a meandering jet on bulk-region and near-wall vortex pairs. A phenomenological model, called the ejection-jet cycle, reveals the relationship between the spatial inner structure of the interface and the invading speed. This model gives insight on the inner-outer interaction in wall-turbulence.

Symmetry reduction in high dimensions, illustrated in a turbulent pipe

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e., for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied one-dimensional PDEs, to a spatially three-dimensional fluid flow, and show that it is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.

Solitary solutions including spatially localized chaos and their interactions in two-dimensional Kolmogorov flow

Two-dimensional Kolmogorov flow in wide periodic boxes is numerically investigated. It is shown that the total flow rate in the direction perpendicular to the force controls the characteristics of the flow, especially the existence of spatially localized solitary solutions such as traveling waves, periodic solutions, and chaotic solutions, which can behave as elementary components of the flow. We propose a procedure to construct approximate solutions consisting of solitary solutions. It is confirmed by direct numerical simulations that these solutions are stable and represent interactions between elementary components such as collisions, coexistence, and collapse of chaos.

Sudden Relaminarization and Lifetimes in Forced Isotropic Turbulence

We demonstrate an unexpected connection between isotropic turbulence and wall-bounded shear flows. We perform direct numerical simulations of isotropic turbulence forced at large scales at moderate Reynolds numbers and observe sudden transitions from a chaotic dynamics to a spatially simple flow, analogous to the laminar state in wall bounded shear flows. We find that the survival probabilities of turbulence are exponential and the typical lifetimes increase superexponentially with the Reynolds number. Our results suggest that both isotropic turbulence and wall-bounded shear flows qualitatively share the same phase-space dynamics.

Study of the instability of the Poiseuille flow using a thermodynamic formalism

The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier-Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec ≈ 2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

Effect of plate permeability on nonlinear stability of the asymptotic suction boundary layer

The nonlinear stability of the asymptotic suction boundary layer is studied numerically, searching for finite-amplitude solutions that bifurcate from the laminar flow state. By changing the boundary conditions for disturbances at the plate from the classical no-slip condition to more physically sound ones, the stability characteristics of the flow may change radically, both for the linearized as well as the nonlinear problem. The wall boundary condition takes into account the permeability K̂ of the plate; for very low permeability, it is acceptable to impose the classical boundary condition (K̂=0). This leads to a Reynolds number of approximately Re(c)=54400 for the onset of linearly unstable waves, and close to Re(g)=3200 for the emergence of nonlinear solutions [F. A. Milinazzo and P. G. Saffman, J. Fluid Mech. 160, 281 (1985); J. H. M. Fransson, Ph.D. thesis, Royal Institute of Technology, KTH, Sweden, 2003]. However, for larger values of the plate's permeability, the lower limit for the existence of linear and nonlinear solutions shifts to significantly lower Reynolds numbers. For the largest permeability studied here, the limit values of the Reynolds numbers reduce down to Re(c)=796 and Re(g)=294. For all cases studied, the solutions bifurcate subcritically toward lower Re, and this leads to the conjecture that they may be involved in the very first stages of a transition scenario similar to the classical route of the Blasius boundary layer initiated by Tollmien-Schlichting (TS) waves. The stability of these nonlinear solutions is also investigated, showing a low-frequency main unstable mode whose growth rate decreases with increasing permeability and with the Reynolds number, following a power law Re(-ρ), where the value of ρ depends on the permeability coefficient K̂. The nonlinear dynamics of the flow in the vicinity of the computed finite-amplitude solutions is finally investigated by direct numerical simulations, providing a viable scenario for subcritical transition due to TS waves.

Coherent structures in wall-bounded turbulence

The inherent difficulty of understanding turbulence has led to researchers attacking the topic in many different ways over the years of turbulence research. Some approaches have been more successful than others, but most only deal with part of the problem. One approach that has seen reasonable success (or at least popularity) is that of attempting to deconstruct the complex and disorganised turbulent flow field into to a set of motions that are in some way organised. These motions are generally called "coherent structures". There are several strands to this approach, from identifying the coherent structures within the flow, defining their characteristics, explaining how they are created, sustained and destroyed, to utilising their features to model the turbulent flow. This review considers research on coherent structures in wall-bounded turbulent flows: a class of flow which is extremely interesting to many scientists (mainly, but not exclusively, physicists and engineers) due to their prevalence in nature, industry and everyday life. This area has seen a lot of activity, particularly in recent years, much of which has been driven by advances in experimental and computational techniques. However, several ideas, developed many years ago based on flow visualisation and intuition, are still both informative and relevant. Indeed, much of the more recent research is firmly indebted to some of the early pioneers of the coherent structures approach. Therefore, in this review, selected historical research is discussed along with the more contemporary advances in an attempt to provide the reader with a good overview of how the field has developed and to highlight the perspicacity of some of the early researchers, as well as providing an overview of our current understanding of the role of coherent structures in wall-bounded turbulent flows.

Localization in a spanwise-extended model of plane Couette flow

We consider a nine-partial-differential-equation (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localization in detail. Of the many steady Eckhaus (spanwise modulational) instabilities identified of global steady states, none lead to a localized state. Spatially localized, time-periodic solutions were found instead, which arise in saddle node bifurcations in the Reynolds number. These solutions appear global (domain filling) in narrow (small spanwise) domains yet can be smoothly continued out to fully spanwise-localized states in very wide domains. This smooth localization behavior, which has also been seen in fully resolved duct flow (S. Okino, Ph.D. thesis, Kyoto University, Kyoto, 2011), indicates that an apparently global flow structure does not have to suffer a modulational instability to localize in wide domains.

Crisis bifurcations in plane Poiseuille flow

Many shear flows follow a route to turbulence that has striking similarities to bifurcation scenarios in low-dimensional dynamical systems. Among the bifurcations that appear, crisis bifurcations are important because they cause global transitions between open and closed attractors, or indicate drastic increases in the range of the state space that is covered by the dynamics. We here study exterior and interior crisis bifurcations in direct numerical simulations of transitional plane Poiseuille flow in a mirror-symmetric subspace. We trace the state space dynamics from the appearance of the first three-dimensional exact coherent structures to the transition from an attractor to a chaotic saddle in an exterior crisis. For intermediate Reynolds numbers, the attractor undergoes several interior crises, in which new states appear and intermittent behavior can be observed. The bifurcations contribute to increasing the complexity of the dynamics and to a more dense coverage of state space.

Reduction of SO(2) symmetry for spatially extended dynamical systems

Spatially extended systems, such as channel or pipe flows, are often equivariant under continuous symmetry transformations, with each state of the flow having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence. Here we describe the "first Fourier mode slice," a very simple, easy to implement reduction of SO(2) symmetry. While the method exhibits rapid variations in phase velocity whenever the magnitude of the first Fourier mode is nearly vanishing, these near singularities can be regularized by a time-scaling transformation. We show that after application of the method, hitherto unseen global structures, for example, Kuramoto-Sivashinsky relative periodic orbits and unstable manifolds of traveling waves, are uncovered.

Distinct organizational States of fully developed turbulent pipe flow

Organizational states of turbulence are identified through novel analysis of large scale pipe flow experiments at a Reynolds number of 35 000. The distinct states are revealed by an azimuthal decomposition of the two-point spatial correlation of the streamwise velocity fluctuation. States with dominant azimuthal wave numbers corresponding to k_{θ}=2,3,4,5,6 are discovered and their structure revealed as a series of alternately rotating quasistreamwise vortices. Such organizational states are highly reminiscent of the nonlinear traveling wave solutions previously identified at Reynolds numbers an order of magnitude lower.

Damping filter method for obtaining spatially localized solutions

Spatially localized structures are key components of turbulence and other spatiotemporally chaotic systems. From a dynamical systems viewpoint, it is desirable to obtain corresponding exact solutions, though their existence is not guaranteed. A damping filter method is introduced to obtain variously localized solutions and adapted in two typical cases. This method introduces a spatially selective damping effect to make a good guess at the exact solution, and we can obtain an exact solution through a continuation with the damping amplitude. The first target is a steady solution to the Swift-Hohenberg equation, which is a representative of bistable systems in which localized solutions coexist and a model for spanwise-localized cases. Not only solutions belonging to the well-known snaking branches but also those belonging to isolated branches known as "isolas" are found with continuation paths between them in phase space extended with the damping amplitude. This indicates that this spatially selective excitation mechanism has an advantage in searching spatially localized solutions. The second target is a spatially localized traveling-wave solution to the Kuramoto-Sivashinsky equation, which is a model for streamwise-localized cases. Since the spatially selective damping effect breaks Galilean and translational invariances, the propagation velocity cannot be determined uniquely while the damping is active, and a singularity arises when these invariances are recovered. We demonstrate that this singularity can be avoided by imposing a simple condition, and a localized traveling-wave solution is obtained with a specific propagation speed.

Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow

The aim in the dynamical systems approach to transitional turbulence is to construct a scaffold in phase space for the dynamics using simple invariant sets (exact solutions) and their stable and unstable manifolds. In large (realistic) domains where turbulence can coexist with laminar flow, this requires identifying exact localized solutions. In wall-bounded shear flows, the first of these has recently been found in pipe flow, but questions remain as to how they are connected to the many known streamwise-periodic solutions. Here we demonstrate that the origin of the first localized solution is in a modulational symmetry-breaking Hopf bifurcation from a known global traveling wave that has twofold rotational symmetry about the pipe axis. Similar behavior is found for a global wave of threefold rotational symmetry, this time leading to two localized relative periodic orbits. The clear implication is that many global solutions should be expected to lead to more realistic localized counterparts through such bifurcations, which provides a constructive route for their generation.

Long-wavelength instability of coherent structures in plane Couette flow

We study the stability of coherent structures in plane Couette flow against long-wavelength perturbations in wide domains that cover several pairs of coherent structures. For one and two pairs of vortices, the states retain the stability properties of the small domains, but for three pairs new unstable modes are found. They are shown to be connected to bifurcations that break the translational symmetry and drive the coherent structures from the spanwise extended state to a modulated one that is a precursor to spanwise localized states. Tracking the stability of the orbits as functions of the spanwise wave length reveals a rich variety of additional bifurcations.

Unstable flow structures in the Blasius boundary layer

Finite amplitude coherent structures with a reflection symmetry in the spanwise direction of a parallel boundary layer flow are reported together with a preliminary analysis of their stability. The search for the solutions is based on the self-sustaining process originally described by Waleffe (Phys. Fluids 9, 883 (1997)). This requires adding a body force to the Navier-Stokes equations; to locate a relevant nonlinear solution it is necessary to perform a continuation in the nonlinear regime and parameter space in order to render the body force of vanishing amplitude. Some states computed display a spanwise spacing between streaks of the same length scale as turbulence flow structures observed in experiments (S.K. Robinson, Ann. Rev. Fluid Mech. 23, 601 (1991)), and are found to be situated within the buffer layer. The exact coherent structures are unstable to small amplitude perturbations and thus may be part of a set of unstable nonlinear states of possible use to describe the turbulent transition. The nonlinear solutions survive down to a displacement thickness Reynolds number Re * = 496 , displaying a 4-vortex structure and an amplitude of the streamwise root-mean-square velocity of 6% scaled with the free-stream velocity. At this Re* the exact coherent structure bifurcates supercritically and this is the point where the laminar Blasius flow starts to cohabit the phase space with alternative simple exact solutions of the Navier-Stokes equations.

Complexity of localised coherent structures in a boundary-layer flow

We study numerically transitional coherent structures in a boundary-layer flow with homogeneous suction at the wall (the so-called asymptotic suction boundary layer ASBL). The dynamics restricted to the laminar-turbulent separatrix is investigated in a spanwise-extended domain that allows for robust localisation of all edge states. We work at fixed Reynolds number and study the edge states as a function of the streamwise period. We demonstrate the complex spatio-temporal dynamics of these localised states, which exhibits multistability and undergoes complex bifurcations leading from periodic to chaotic regimes. It is argued that in all regimes the dynamics restricted to the edge is essentially low-dimensional and non-extensive.

Increasing lifetimes and the growing saddles of shear flow turbulence

In linearly stable shear flows, turbulence spontaneously decays with a characteristic lifetime that varies with Reynolds number. The lifetime sharply increases with Reynolds number so that a possible divergence marking the transition to sustained turbulence at a critical point has been discussed. We present a mechanism by which the lifetimes increase: in the system's state space, turbulent motion is supported by a chaotic saddle. Inside this saddle a locally attracting periodic orbit is created and undergoes a traditional bifurcation sequence generating chaos. The formed new "turbulent bubble" is initially an attractor supporting persistent chaotic dynamics. Soon after its creation, it collides with its own boundary, by which it becomes leaky and dynamically connected with the surrounding structures. The complexity of the chaotic saddle that supports transient turbulence hence increases by incorporating the remnant of a new bubble. As a a result, the time it takes for a trajectory to leave the saddle and decay to the laminar state is increased. We demonstrate this phenomenon in plane Couette flow and show that characteristic lifetimes vary nonsmoothly and nonmonotonically with Reynolds number.

Spatial localization in heterogeneous systems

We study spatial localization in the generalized Swift-Hohenberg equation with either quadratic-cubic or cubic-quintic nonlinearity subject to spatially heterogeneous forcing. Different types of forcing (sinusoidal or Gaussian) with different spatial scales are considered and the corresponding localized snaking structures are computed. The results indicate that spatial heterogeneity exerts a significant influence on the location of spatially localized structures in both parameter space and physical space, and on their stability properties. The results are expected to assist in the interpretation of experiments on localized structures where departures from spatial homogeneity are generally unavoidable.

Nature of laminar-turbulence intermittency in shear flows

In pipe, channel, and boundary layer flows turbulence first occurs intermittently in space and time: at moderate Reynolds numbers domains of disordered turbulent motion are separated by quiescent laminar regions. Based on direct numerical simulations of pipe flow we argue here that the spatial intermittency has its origin in a nearest neighbor interaction between turbulent regions. We further show that in this regime turbulent flows are intrinsically intermittent with a well-defined equilibrium turbulent fraction but without ever assuming a steady pattern. This transition scenario is analogous to that found in simple models such as coupled map lattices. The scaling observed implies that laminar intermissions of the turbulent flow will persist to arbitrarily large Reynolds numbers.