Turbulence is the major cause of friction losses in transport processes and it is responsible for a drastic drag increase in flows over bounding surfaces. While much effort is invested into developing ways to control and reduce turbulence intensities [1][2][3] , so far no methods exist to altogether eliminate turbulence if velocities are sufficiently large. We demonstrate for pipe flow that appropriate distortions to the velocity profile lead to a complete collapse of turbulence and subsequently friction losses are reduced by as much as 90%. Counterintuitively, the return to laminar motion is accomplished by initially increasing turbulence intensities or by transiently amplifying wall shear. Since neither the Reynolds number nor the shear stresses decrease (the latter often increase), these measures are not indicative of turbulence collapse. Instead, an amplification mechanism 4,5 measuring the interaction between eddies and the mean shear is found to set a threshold below which turbulence is suppressed beyond recovery.Flows through pipes and hydraulic networks are generally turbulent and the friction losses encountered in these flows are responsible for approximately 10% of the global electric energy consumption. Here turbulence causes a severe drag increase and consequently much larger forces are needed to maintain desired flow rates. In pipes, both laminar and turbulent states are stable (the former is believed to be linearly stable for all Reynolds number (Re) values; the latter is stable if Re > 2, 040 (ref. 6 )), but with increasing speed the laminar state becomes more and more susceptible to small disturbances. Hence, in practice most flows are turbulent at sufficiently large Re. While the stability of laminar flow has been studied in great detail, little attention has been paid to the susceptibility of turbulence, the general assumption being that once turbulence is established it is stable.Many turbulence control strategies have been put forward to reduce the drag encountered in shear flows [7][8][9][10][11][12][13][14][15][16][17] . Recent strategies employ feedback mechanisms to actively counter selected velocity components or vortices. Such methods usually require knowledge of the full turbulent velocity field. In computer simulations 7,8 , it could be demonstrated that under these ideal conditions, flows at a low Re number can even be relaminarized. In experiments, the required detailed manipulation of the time-dependent velocity field is, however, currently impossible to achieve. Other studies employ passive (for example, riblets) or active (oscillations or excitation of travelling waves) methods to interfere with the near-wall turbulence creation. Typically here drag reduction of 10 to 40% has been reported, but often the control cost is substantially higher than the gain, or a net gain can be achieved only in a narrow Re number regime.Instead of attempting to control or counter certain components of the complex fluctuating flow fields, we will show in the following that by appropriately dist...

Chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this also be the case for the infinite-dimensional dynamics of Navier-Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics-specifically, whether periodic orbits play the same role in highdimensional nonlinear systems like the Navier-Stokes equations as they do in lowerdimensional systems-is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at Re = 2500, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

During bacterial cell division, the tubulin-homolog FtsZ forms a ring-like structure at the center of the cell. This Z-ring not only organizes the division machinery, but treadmilling of FtsZ filaments was also found to play a key role in distributing proteins at the division site. What regulates the architecture, dynamics and stability of the Z-ring is currently unknown, but FtsZ-associated proteins are known to play an important role. Here, using an in vitro reconstitution approach, we studied how the well-conserved protein ZapA affects FtsZ treadmilling and filament organization into large-scale patterns. Using high-resolution fluorescence microscopy and quantitative image analysis, we found that ZapA cooperatively increases the spatial order of the filament network, but binds only transiently to FtsZ filaments and has no effect on filament length and treadmilling velocity. Together, our data provides a model for how FtsZ-associated proteins can increase the precision and stability of the bacterial cell division machinery in a switch-like manner.

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.

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In shear flows at transitional Reynolds numbers, localized patches of turbulence, known as puffs, coexist with the laminar flow. Recently, Avila et al., Phys. Rev. Let. 110, 224502 (2013) discovered two spatially localized relative periodic solutions for pipe flow, which appeared in a saddle-node bifurcation at low speeds. Combining slicing methods for continuous symmetry reduction with Poincaré sections for the first time in a shear flow setting, we compute and visualize the unstable manifold of the lower-branch solution and show that it contains a heteroclinic connection to the upper branch solution. Surprisingly this connection even persists far above the bifurcation point and appears to mediate puff generation, providing a dynamical understanding of this phenomenon.

Over the past decade, the edge of chaos has proven to be a fruitful starting point for investigations of shear flows when the laminar base flow is linearly stable. Numerous computational studies of shear flows demonstrated the existence of states that separate laminar and turbulent regions of the state space. In addition, some studies determined invariant solutions that reside on this edge. In this paper, we study the unstable manifold of one such solution with the aid of continuous symmetry-reduction, which we formulate here for the first time for the simultaneous quotiening of axial and azimuthal symmetries. Upon our investigation of the unstable manifold, we discover a previously unknown traveling wave solution on the laminar-turbulent boundary with a relatively complex structure. By means of low-dimensional projections, we visualize different dynamical paths that connect these solutions to the turbulence. Our numerical experiments demonstrate that the laminar-turbulent boundary exhibits qualitatively different regions whose properties are influenced by the nearby invariant solutions.

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