Shear flows undergo a sudden transition from laminar to turbulent motion as the velocity increases, and the onset of turbulence radically changes transport efficiency and mixing properties. Even for the well-studied case of pipe flow, it has not been possible to determine at what Reynolds number the motion will be either persistently turbulent or ultimately laminar. We show that in pipes, turbulence that is transient at low Reynolds numbers becomes sustained at a distinct critical point. Through extensive experiments and computer simulations, we were able to identify and characterize the processes ultimately responsible for sustaining turbulence. In contrast to the classical Landau-Ruelle-Takens view that turbulence arises from an increase in the temporal complexity of fluid motion, here, spatial proliferation of chaotic domains is the decisive process and intrinsic to the nature of fluid turbulence.

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.

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.

Turbulence is ubiquitous in nature yet even for the case of ordinary Newtonian fluids like water our understanding of this phenomenon is limited. Many liquids of practical importance however are more complicated (e.g. blood, polymer melts or paints), they exhibit elastic as well as viscous characteristics and the relation between stress and strain is nonlinear. We here demonstrate for a model system of such complex fluids that at high shear rates turbulence is not simply modified as previously believed but it is suppressed and replaced by a new type of disordered motion, elasto-inertial turbulence (EIT). EIT is found to occur at much lower Reynolds numbers than Newtonian turbulence and the dynamical properties differ significantly. In particular the drag is strongly reduced and the observed friction scaling resolves a longstanding puzzle in non-Newtonian fluid mechanics regarding the nature of the so-called maximum drag reduction asymptote. Theoretical considerations imply that EIT will arise in complex fluids if the extensional viscosity is sufficiently large

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.

We report the results of an experimental investigation of the transition to turbulence in a pipe over approximately an order of magnitude range in the Reynolds number Re. A novel scaling law is uncovered using a systematic experimental procedure which permits contact to be made with modern theoretical thinking. The principal result we uncover is a scaling law which indicates that the amplitude of perturbation required to cause transition scales as O(Re-1).

This manuscript is the original submission to Nature. The final (published) version can be accessed at http://www.nature.com/nature/journal/v526/n7574/full/nature15701.html 1 arXiv:1510.09143v1 [physics.flu-dyn] Oct 2015Over a century of research into the origin of turbulence in wallbounded shear flows has resulted in a puzzling picture in which turbulence appears in a variety of different states competing with laminar background flow. [1][2][3][4][5][6] At slightly higher speeds the situation changes distinctly and the entire flow is turbulent. Neither the origin of the different states encountered during transition, nor their front dynamics, let alone the transformation to full turbulence could be explained to date. Combining experiments, theory and computer simulations here we uncover the bifurcation scenario organising the route to fully turbulent pipe flow and explain the front dynamics of the different states encountered in the process. Key to resolving this problem is the interpretation of the flow as a bistable system with nonlinear propagation (advection) of turbulent fronts. These findings bridge the gap between our understanding of the onset of turbulence 7 and fully turbulent flows. 8,9 The sudden appearance of localised turbulent patches in an otherwise quiescent flow was first observed by Osborne Reynolds for pipe flow 1 and has since been found to be the starting point of turbulence in most shear flows. 2,4,[10][11][12][13][14][15] Curiously, in this regime it is impossible to maintain turbulence over extended regions as it automatically 16,17 reduces to patches of characteristic size, called puffs in pipe flow (see Fig. 1a). Puffs can decay, or else split and thereby multiply. Once the Reynolds number R > 2040 the splitting process outweighs decay, resulting in sustained disordered motion. 7 Although sustained, turbulence at these low R only consists of puffs surrounded by laminar flow (Fig. 1a) and cannot form larger clusters. 17,18 At larger flow rates, the situation is fundamentally different: once triggered, turbulence aggressively expands and eliminates all laminar motion (Fig. 1b). Fully turbulent flow is now the natural state of the system and only then do wallbounded shear flows have characteristic mean properties such as the Blasius or Prandtl-von Karman friction laws. 9 This rise of fully turbulent flow has remained unexplained despite the fact that this transformation occurs in virtually all shear flows and generally dominates the dynamics at sufficiently large Reynolds numbers.A classical diagnostic for the formation of turbulence 2-5, 19, 20 is the propagation speed of the In each case, the flow is initially seeded with localised turbulent patches and the subsequent evolution is visualised via space-time plots in a reference frame co-moving with structures. Colours indicate the value of u 2 r + u 2 θ . Further highlighting the distinction between cases, shown at the top are cross sections of instantaneous flow within the pipe. A 35D section is shown with the vertical direction str...

The collapse of turbulence, observable in shear flows at low Reynolds numbers, raises the question if turbulence is generically of a transient nature or becomes sustained at some critical point. Recent data have led to conflicting views with the majority of studies supporting the model of turbulence turning into an attracting state. Here we present lifetime measurements of turbulence in pipe flow spanning 8 orders of magnitude in time, drastically extending all previous investigations. We show that no critical point exists in this regime and that in contrast to the prevailing view the turbulent state remains transient. To our knowledge this is the first observation of superexponential transients in turbulence, confirming a conjecture derived from low-dimensional systems. [4][5][6]. Surprisingly, at relatively low Reynolds numbers (Re & 2000) the turbulent state is not stable and after long times suddenly collapses [7][8][9][10][11][12]. This behavior is reminiscent of memoryless processes in nonlinear systems. In phase space the dynamics can be described by a complex structure giving rise to the disordered dynamics, a socalled chaotic repeller [13]. Underlying such a structure are unstable states and for pipe flow unstable solutions to the governing equations have been identified in the form of traveling waves [14,15]. Surprisingly clear transients of such traveling waves were observed in experiments [16,17] confirming their relevance to the turbulent dynamics. More recently traveling wave transients were also reported in numerical studies [18,19].A way to probe the validity of this model is to measure the lifetime of turbulence in the transient regime. Previous experimental and numerical lifetime measurements have shown approximately exponential probability distributions [8,10,11,20,21] which suggests that the probability for a turbulent structure to decay is independent of its age and hence that this process is memoryless as would be expected for the escape from a chaotic saddle. Here the probability for a flow to still be turbulent after a time t at a fixed Reynolds number (Re) is then given bywhere is the characteristic lifetime ( À1 can be also interpreted as the escape rate) and t 0 is the initial time period required for turbulence to form after the disturbance has been applied to the laminar flow at t ¼ 0. an infinite lifetime is only reached in the asymptotic limit Re ! 1. Subsequently a number of studies have questioned this finding and again entertained the occurrence of a boundary crisis [11,22,23]. A clear constraint of all previous investigations is the limited range in lifetimes measured. Typically scaling laws were postulated from data covering 2 orders of magnitude. Numerical simulations are particularly problematic because in order to capture the quantitatively correct behavior computations have to be carried out in large domains, which severely restricts the number of realizations N that are manageable (N < 50) [11]. Consequently the statistics are often insufficiently resolved resulting in a...