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.

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...

Although the equations governing fluid flow are well known, there are no analytical expressions that describe the complexity of turbulent motion. A recent proposition is that in analogy to low dimensional chaotic systems, turbulence is organized around unstable solutions of the governing equations which provide the building blocks of the disordered dynamics. We report the discovery of periodic solutions which just like intermittent turbulence are spatially localized and show that turbulent transients arise from one such solution branch.

Flows through pipes and channels are the most common means to transport fluids in practical applications and equally occur in numerous natural systems. In general, the transfer of fluids is energetically far more efficient if the motion is smooth and laminar because the friction losses are lower. However, even at moderate velocities pipe and channel flows are sensitive to minute disturbances, and in practice most flows are turbulent. Investigating the motion and spatial distribution of vortices, we uncovered an amplification mechanism that constantly feeds energy from the mean shear into turbulent eddies. At intermediate flow rates, a simple control mechanism suffices to intercept this energy transfer by reducing inflection points in the velocity profile. When activated, an immediate collapse of turbulence is observed, and the flow relaminarizes.

The onset of shear flow turbulence is characterized by turbulent patches bounded by regions of laminar flow. At low Reynolds numbers localized turbulence relaminarises, raising the question of whether it is transient in nature or it becomes sustained at a critical threshold. We present extensive numerical simulations and a detailed statistical analysis of the data, in order to shed light on the sources of the discrepancies present in the literature. The results are in excellent quantitative agreement with recent experiments and show that the turbulent lifetimes increase super-exponentially with Reynolds number. In addition, we provide evidence for a lower bound below which there are no meta-stable characteristics of the transients, i.e. the relaminarisation process is no longer memoryless.Comment: 10 pages, 4 figures, J. Fluid Mech. Revised text and figure to improve presentation and clarity. Few new simulations include