Pipe flow is a prominent example among the shear flows that undergo transition to turbulence without mediation by a linear instability of the laminar profile. Experiments on pipe flow, as well as plane Couette and plane Poiseuille flow, show that triggering turbulence depends sensitively on initial conditions, that between the laminar and the turbulent states there exists no intermediate state with simple spatial or temporal characteristics, and that turbulence is not persistent, i.e., it can decay again, if the observation time is long enough. All these features can consistently be explained on the assumption that the turbulent state corresponds to a chaotic saddle in state space. The goal of this review is to explain this concept, summarize the numerical and experimental evidence for pipe flow, and outline the consequences for related flows.

Turbulent Taylor-Couette flow with arbitrary rotation frequencies ω 1 , ω 2 of the two coaxial cylinders with radii r 1 < r 2 is analysed theoretically. The current J ω of the angular velocity ω(x, t) = u ϕ (r, ϕ, z, t)/r across the cylinder gap and and the excess energy dissipation rate ε w due to the turbulent, convective fluctuations (the 'wind') are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor-Couette flow with thermal Rayleigh-Bénard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio η = r 1 /r 2 or the gap width d = r 2 − r 1 between the cylinders. A quantity σ corresponding to the Prandtl number in Rayleigh-Bénard flow can be introduced, σ = ((1 + η)/2)/ √ η) 4 . In Taylor-Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh-Bénard flow. The analogue of the Rayleigh number is the Taylor number, defined as T a ∝ (ω 1 − ω 2 ) 2 times a specific geometrical factor. The experimental data show no pure power law, but the exponent α of the torque versus the rotation frequency ω 1 depends on the driving frequency ω 1 . An explanation for the physical origin of the ω 1 -dependence of the measured local power-law exponents α(ω 1 ) is put forward. Also, the dependence of the torque on the gap width η is discussed and, in particular its strong increase for η → 1.

Transition to turbulence in pipe flow is one of the most fundamental and longest-standing problems in fluid dynamics. Stability theory suggests that the flow remains laminar for all flow rates, but in practice pipe flow becomes turbulent even at moderate speeds. This transition drastically affects the transport efficiency of mass, momentum, and heat. On the basis of the recent discovery of unstable traveling waves in computational studies of the Navier-Stokes equations and ideas from dynamical systems theory, a model for the transition process has been suggested. We report experimental observation of these traveling waves in pipe flow, confirming the proposed transition scenario and suggesting that the dynamics associated with these unstable states may indeed capture the nature of fluid turbulence.

Synchronization and wave formation in one-dimensional ciliary arrays are studied analytically and numerically. We develop a simple model for ciliary motion that is complex enough to describe well the behavior of beating cilia but simple enough to study collective effects analytically. Beating cilia are described as phase oscillators moving on circular trajectories with a variable radius. This radial degree of freedom turns out to be essential for the occurrence of hydrodynamically induced synchronization of ciliary beating between neighboring cilia. The transitions to the synchronized and phase-locked state of two cilia and the formation of metachronal waves in ciliary chains with different boundary conditions are discussed.

We study the transition between laminar and turbulent states in a Galerkin representation of a parallel shear flow, where a stable laminar flow and a transient turbulent flow state coexist. The regions of initial conditions where the lifetimes show strong fluctuations and a sensitive dependence on initial conditions are separated from the ones with a smooth variation of lifetimes by an object in phase space which we call the "edge of chaos." We describe techniques to identify and follow the edge, and our results indicate that the edge is a surface. For low Reynolds numbers we find that the surface coincides with the stable manifold of a periodic orbit, whereas at higher Reynolds numbers it is the stable set of a higher-dimensional chaotic object.

A family of three-dimensional travelling waves for flow through a pipe of circular cross section is identified. The travelling waves are dominated by pairs of downstream vortices and streaks. They originate in saddle-node bifurcations at Reynolds numbers as low as 1250. All states are immediately unstable. Their dynamical significance is that they provide a skeleton for the formation of a chaotic saddle that can explain the intermittent transition to turbulence and the sensitive dependence on initial conditions in this shear flow.PACS numbers: 47.20.Ft ,47.20.Lz ,47.35.+i Based on decades of studies it is consensus that HagenPoiseuille flow through a pipe of circular cross section belongs to the class of shear flows that does not become linearly unstable. Nevertheless, it undergoes an intermittent transition to turbulence for sufficiently high Reynolds numbers and sufficiently large initial perturbations, as first documented in the classic experiments by Reynolds [1]. Since then many studies have analyzed the mechanisms of transition and the properties of the turbulent state [2,3,4,5,6]. Particularly relevant to the present analysis are the investigations by Darbyshire and Mullin [7] which clearly show a strong sensitivity to perturbations and a broad intermittent range of decaying and turbulent perturbations in an amplitude vs. Reynolds number plane. Experimental and numerical studies [3,8] show that the turbulent flow in the transition region is dominated by downstream vortices and streaks. Various models for their dynamics have been analyzed [9,10]. Taking the full nonlinearity into account Waleffe developed the concept of a nonlinear turbulence cycle for the regeneration of vortices and streaks [11]. In addition, stationary solutions and travelling waves have been found in the full nonlinear equations for plane Couette, Taylor-Couette and plane Poiseuille flow [12,13,14,15,16]. It has been suggested that these structures provide a skeleton for the transition to turbulence and the observed intermittency [17,18]. They clearly dominate various observables in low Reynolds number turbulent flows [10], and are also relevant for an understanding of the effects of non-Newtonian additives [19].The existence of exact coherent states in pipe flow has been an object of speculation for some time [5,6,7]. As we will show here Hagen-Poiseuille flow supports families of travelling waves with structures similar to those observed in other shear flows as well. This underlines the significance of vortex-streak interactions also in this system and opens alternative routes to modelling and controlling pipe flow. * Electronic address: Holger.Faisst@physik.uni-marburg.deThe existence of stationary states in plane Couette flow is connected with an inversion symmetry in the laminar profile. In the absence of such a symmetry in pipe flow the simplest states we can expect are travelling waves (TWs), i.e. coherent structures that move with constant wave speed. The wave speed depends on shape and structure and is not known in advance....

Generally, the motion of fluids is smooth and laminar at low speeds but becomes highly disordered and turbulent as the velocity increases. The transition from laminar to turbulent flow can involve a sequence of instabilities in which the system realizes progressively more complicated states, or it can occur suddenly. Once the transition has taken place, it is generally assumed that, under steady conditions, the turbulent state will persist indefinitely. The flow of a fluid down a straight pipe provides a ubiquitous example of a shear flow undergoing a sudden transition from laminar to turbulent motion. Extensive calculations and experimental studies have shown that, at relatively low flow rates, turbulence in pipes is transient, and is characterized by an exponential distribution of lifetimes. They also suggest that for Reynolds numbers exceeding a critical value the lifetime diverges (that is, becomes infinitely large), marking a change from transient to persistent turbulence. Here we present experimental data and numerical calculations covering more than two decades of lifetimes, showing that the lifetime does not in fact diverge but rather increases exponentially with the Reynolds number. This implies that turbulence in pipes is only a transient event (contrary to the commonly accepted view), and that the turbulent and laminar states remain dynamically connected, suggesting avenues for turbulence control.

Soon after the crowd streamed on to London's Millennium Bridge on the day it opened, the bridge started to sway from side to side: many pedestrians fell spontaneously into step with the bridge's vibrations, inadvertently amplifying them. Here we model this unexpected and now notorious phenomenon--which was not due to the bridge's innovative design as was first thought--by adapting ideas originally developed to describe the collective synchronization of biological oscillators such as neurons and fireflies. Our approach should help engineers to estimate the damping needed to stabilize other exceptionally crowded footbridges against synchronous lateral excitation by pedestrians.