@article{14341, abstract = {Flows through pipes and channels are, in practice, almost always turbulent, and the multiscale eddying motion is responsible for a major part of the encountered friction losses and pumping costs1. Conversely, for pulsatile flows, in particular for aortic blood flow, turbulence levels remain low despite relatively large peak velocities. For aortic blood flow, high turbulence levels are intolerable as they would damage the shear-sensitive endothelial cell layer2,3,4,5. Here we show that turbulence in ordinary pipe flow is diminished if the flow is driven in a pulsatile mode that incorporates all the key features of the cardiac waveform. At Reynolds numbers comparable to those of aortic blood flow, turbulence is largely inhibited, whereas at much higher speeds, the turbulent drag is reduced by more than 25%. This specific operation mode is more efficient when compared with steady driving, which is the present situation for virtually all fluid transport processes ranging from heating circuits to water, gas and oil pipelines.}, author = {Scarselli, Davide and Lopez Alonso, Jose M and Varshney, Atul and Hof, Björn}, issn = {1476-4687}, journal = {Nature}, number = {7977}, pages = {71--74}, publisher = {Springer Nature}, title = {{Turbulence suppression by cardiac-cycle-inspired driving of pipe flow}}, doi = {10.1038/s41586-023-06399-5}, volume = {621}, year = {2023}, } @article{14361, abstract = {Whether one considers swarming insects, flocking birds, or bacterial colonies, collective motion arises from the coordination of individuals and entails the adjustment of their respective velocities. In particular, in close confinements, such as those encountered by dense cell populations during development or regeneration, collective migration can only arise coordinately. Yet, how individuals unify their velocities is often not understood. Focusing on a finite number of cells in circular confinements, we identify waves of polymerizing actin that function as a pacemaker governing the speed of individual cells. We show that the onset of collective motion coincides with the synchronization of the wave nucleation frequencies across the population. Employing a simpler and more readily accessible mechanical model system of active spheres, we identify the synchronization of the individuals’ internal oscillators as one of the essential requirements to reach the corresponding collective state. The mechanical ‘toy’ experiment illustrates that the global synchronous state is achieved by nearest neighbor coupling. We suggest by analogy that local coupling and the synchronization of actin waves are essential for the emergent, self-organized motion of cell collectives.}, author = {Riedl, Michael and Mayer, Isabelle D and Merrin, Jack and Sixt, Michael K and Hof, Björn}, issn = {2041-1723}, journal = {Nature Communications}, publisher = {Springer Nature}, title = {{Synchronization in collectively moving inanimate and living active matter}}, doi = {10.1038/s41467-023-41432-1}, volume = {14}, year = {2023}, } @article{14466, abstract = {The first long-lived turbulent structures observable in planar shear flows take the form of localized stripes, inclined with respect to the mean flow direction. The dynamics of these stripes is central to transition, and recent studies proposed an analogy to directed percolation where the stripes’ proliferation is ultimately responsible for the turbulence becoming sustained. In the present study we focus on the internal stripe dynamics as well as on the eventual stripe expansion, and we compare the underlying mechanisms in pressure- and shear-driven planar flows, respectively, plane-Poiseuille and plane-Couette flow. Despite the similarities of the overall laminar–turbulence patterns, the stripe proliferation processes in the two cases are fundamentally different. Starting from the growth and sustenance of individual stripes, we find that in plane-Couette flow new streaks are created stochastically throughout the stripe whereas in plane-Poiseuille flow streak creation is deterministic and occurs locally at the downstream tip. Because of the up/downstream symmetry, Couette stripes, in contrast to Poiseuille stripes, have two weak and two strong laminar turbulent interfaces. These differences in symmetry as well as in internal growth give rise to two fundamentally different stripe splitting mechanisms. In plane-Poiseuille flow splitting is connected to the elongational growth of the original stripe, and it results from a break-off/shedding of the stripe's tail. In plane-Couette flow splitting follows from a broadening of the original stripe and a division along the stripe into two slimmer stripes.}, author = {Marensi, Elena and Yalniz, Gökhan and Hof, Björn}, issn = {1469-7645}, journal = {Journal of Fluid Mechanics}, keywords = {turbulence, transition to turbulence, patterns}, publisher = {Cambridge University Press}, title = {{Dynamics and proliferation of turbulent stripes in plane-Poiseuille and plane-Couette flows}}, doi = {10.1017/jfm.2023.780}, volume = {974}, year = {2023}, } @article{13274, abstract = {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.}, author = {Paranjape, Chaitanya S and Yalniz, Gökhan and Duguet, Yohann and Budanur, Nazmi B and Hof, Björn}, issn = {1079-7114}, journal = {Physical Review Letters}, keywords = {General Physics and Astronomy}, number = {3}, publisher = {American Physical Society}, title = {{Direct path from turbulence to time-periodic solutions}}, doi = {10.1103/physrevlett.131.034002}, volume = {131}, year = {2023}, } @article{12105, abstract = {Data-driven dimensionality reduction methods such as proper orthogonal decomposition and dynamic mode decomposition have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration, as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for fluid flows in rectangular channels by formulating a continuous symmetry reduction method that eliminates the translations in the streamwise and spanwise directions simultaneously. We demonstrate our method by computing the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding windows of data obtained from the transitional plane-Couette and turbulent plane-Poiseuille flow simulations. In the former setting, SRDMD captures the dynamics in the vicinity of the invariant solutions with translation symmetries, i.e. travelling waves and relative periodic orbits, whereas in the latter, our calculations reveal episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.}, author = {Marensi, Elena and Yalniz, Gökhan and Hof, Björn and Budanur, Nazmi B}, issn = {1469-7645}, journal = {Journal of Fluid Mechanics}, publisher = {Cambridge University Press}, title = {{Symmetry-reduced dynamic mode decomposition of near-wall turbulence}}, doi = {10.1017/jfm.2022.1001}, volume = {954}, year = {2023}, } @article{12165, abstract = {It may come as a surprise that a phenomenon as ubiquitous and prominent as the transition from laminar to turbulent flow has resisted combined efforts by physicists, engineers and mathematicians, and remained unresolved for almost one and a half centuries. In recent years, various studies have proposed analogies to directed percolation, a well-known universality class in statistical mechanics, which describes a non-equilibrium phase transition from a fluctuating active phase into an absorbing state. It is this unlikely relation between the multiscale, high-dimensional dynamics that signify the transition process in virtually all flows of practical relevance, and the arguably most basic non-equilibrium phase transition, that so far has mainly been the subject of model studies, which I review in this Perspective.}, author = {Hof, Björn}, issn = {2522-5820}, journal = {Nature Reviews Physics}, keywords = {General Physics and Astronomy}, pages = {62--72}, publisher = {Springer Nature}, title = {{Directed percolation and the transition to turbulence}}, doi = {10.1038/s42254-022-00539-y}, volume = {5}, year = {2023}, } @article{12681, abstract = {The dissolution of minute concentration of polymers in wall-bounded flows is well-known for its unparalleled ability to reduce turbulent friction drag. Another phenomenon, elasto-inertial turbulence (EIT), has been far less studied even though elastic instabilities have already been observed in dilute polymer solutions before the discovery of polymer drag reduction. EIT is a chaotic state driven by polymer dynamics that is observed across many orders of magnitude in Reynolds number. It involves energy transfer from small elastic scales to large flow scales. The investigation of the mechanisms of EIT offers the possibility to better understand other complex phenomena such as elastic turbulence and maximum drag reduction. In this review, we survey recent research efforts that are advancing the understanding of the dynamics of EIT. We highlight the fundamental differences between EIT and Newtonian/inertial turbulence from the perspective of experiments, numerical simulations, instabilities, and coherent structures. Finally, we discuss the possible links between EIT and elastic turbulence and polymer drag reduction, as well as the remaining challenges in unraveling the self-sustaining mechanism of EIT.}, author = {Dubief, Yves and Terrapon, Vincent E. and Hof, Björn}, issn = {1545-4479}, journal = {Annual Review of Fluid Mechanics}, number = {1}, pages = {675--705}, publisher = {Annual Reviews}, title = {{Elasto-inertial turbulence}}, doi = {10.1146/annurev-fluid-032822-025933}, volume = {55}, year = {2023}, } @article{12682, abstract = {Since the seminal studies by Osborne Reynolds in the nineteenth century, pipe flow has served as a primary prototype for investigating the transition to turbulence in wall-bounded flows. Despite the apparent simplicity of this flow, various facets of this problem have occupied researchers for more than a century. Here we review insights from three distinct perspectives: (a) stability and susceptibility of laminar flow, (b) phase transition and spatiotemporal dynamics, and (c) dynamical systems analysis of the Navier—Stokes equations. We show how these perspectives have led to a profound understanding of the onset of turbulence in pipe flow. Outstanding open points, applications to flows of complex fluids, and similarities with other wall-bounded flows are discussed.}, author = {Avila, Marc and Barkley, Dwight and Hof, Björn}, issn = {0066-4189}, journal = {Annual Review of Fluid Mechanics}, pages = {575--602}, publisher = {Annual Reviews}, title = {{Transition to turbulence in pipe flow}}, doi = {10.1146/annurev-fluid-120720-025957}, volume = {55}, year = {2023}, } @article{10791, abstract = {The mammalian neocortex is composed of diverse neuronal and glial cell classes that broadly arrange in six distinct laminae. Cortical layers emerge during development and defects in the developmental programs that orchestrate cortical lamination are associated with neurodevelopmental diseases. The developmental principle of cortical layer formation depends on concerted radial projection neuron migration, from their birthplace to their final target position. Radial migration occurs in defined sequential steps, regulated by a large array of signaling pathways. However, based on genetic loss-of-function experiments, most studies have thus far focused on the role of cell-autonomous gene function. Yet, cortical neuron migration in situ is a complex process and migrating neurons traverse along diverse cellular compartments and environments. The role of tissue-wide properties and genetic state in radial neuron migration is however not clear. Here we utilized mosaic analysis with double markers (MADM) technology to either sparsely or globally delete gene function, followed by quantitative single-cell phenotyping. The MADM-based gene ablation paradigms in combination with computational modeling demonstrated that global tissue-wide effects predominate cell-autonomous gene function albeit in a gene-specific manner. Our results thus suggest that the genetic landscape in a tissue critically affects the overall migration phenotype of individual cortical projection neurons. In a broader context, our findings imply that global tissue-wide effects represent an essential component of the underlying etiology associated with focal malformations of cortical development in particular, and neurological diseases in general.}, author = {Hansen, Andi H and Pauler, Florian and Riedl, Michael and Streicher, Carmen and Heger, Anna-Magdalena and Laukoter, Susanne and Sommer, Christoph M and Nicolas, Armel and Hof, Björn and Tsai, Li Huei and Rülicke, Thomas and Hippenmeyer, Simon}, issn = {2753-149X}, journal = {Oxford Open Neuroscience}, number = {1}, publisher = {Oxford Academic}, title = {{Tissue-wide effects override cell-intrinsic gene function in radial neuron migration}}, doi = {10.1093/oons/kvac009}, volume = {1}, year = {2022}, } @article{11704, abstract = {In Fall 2020, several European countries reported rapid increases in COVID-19 cases along with growing estimates of the effective reproduction rates. Such an acceleration in epidemic spread is usually attributed to time-dependent effects, e.g. human travel, seasonal behavioral changes, mutations of the pathogen etc. In this case however the acceleration occurred when counter measures such as testing and contact tracing exceeded their capacity limit. Considering Austria as an example, here we show that this dynamics can be captured by a time-independent, i.e. autonomous, compartmental model that incorporates these capacity limits. In this model, the epidemic acceleration coincides with the exhaustion of mitigation efforts, resulting in an increasing fraction of undetected cases that drive the effective reproduction rate progressively higher. We demonstrate that standard models which does not include this effect necessarily result in a systematic underestimation of the effective reproduction rate.}, author = {Budanur, Nazmi B and Hof, Björn}, issn = {1932-6203}, journal = {PLoS ONE}, number = {7}, publisher = {Public Library of Science}, title = {{An autonomous compartmental model for accelerating epidemics}}, doi = {10.1371/journal.pone.0269975}, volume = {17}, year = {2022}, } @article{10654, abstract = {Directed percolation (DP) has recently emerged as a possible solution to the century old puzzle surrounding the transition to turbulence. Multiple model studies reported DP exponents, however, experimental evidence is limited since the largest possible observation times are orders of magnitude shorter than the flows’ characteristic timescales. An exception is cylindrical Couette flow where the limit is not temporal, but rather the realizable system size. We present experiments in a Couette setup of unprecedented azimuthal and axial aspect ratios. Approaching the critical point to within less than 0.1% we determine five critical exponents, all of which are in excellent agreement with the 2+1D DP universality class. The complex dynamics encountered at the onset of turbulence can hence be fully rationalized within the framework of statistical mechanics.}, author = {Klotz, Lukasz and Lemoult, Grégoire M and Avila, Kerstin and Hof, Björn}, issn = {1079-7114}, journal = {Physical Review Letters}, number = {1}, publisher = {American Physical Society}, title = {{Phase transition to turbulence in spatially extended shear flows}}, doi = {10.1103/PhysRevLett.128.014502}, volume = {128}, year = {2022}, } @article{12134, abstract = {Standard epidemic models exhibit one continuous, second order phase transition to macroscopic outbreaks. However, interventions to control outbreaks may fundamentally alter epidemic dynamics. Here we reveal how such interventions modify the type of phase transition. In particular, we uncover three distinct types of explosive phase transitions for epidemic dynamics with capacity-limited interventions. Depending on the capacity limit, interventions may (i) leave the standard second order phase transition unchanged but exponentially suppress the probability of large outbreaks, (ii) induce a first-order discontinuous transition to macroscopic outbreaks, or (iii) cause a secondary explosive yet continuous third-order transition. These insights highlight inherent limitations in predicting and containing epidemic outbreaks. More generally our study offers a cornerstone example of a third-order explosive phase transition in complex systems.}, author = {Börner, Georg and Schröder, Malte and Scarselli, Davide and Budanur, Nazmi B and Hof, Björn and Timme, Marc}, issn = {2632-072X}, journal = {Journal of Physics: Complexity}, keywords = {Artificial Intelligence, Computer Networks and Communications, Computer Science Applications, Information Systems}, number = {4}, publisher = {IOP Publishing}, title = {{Explosive transitions in epidemic dynamics}}, doi = {10.1088/2632-072x/ac99cd}, volume = {3}, year = {2022}, } @article{12259, abstract = {Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions. }, author = {Choueiri, George H and Suri, Balachandra and Merrin, Jack and Serbyn, Maksym and Hof, Björn and Budanur, Nazmi B}, issn = {1089-7682}, journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}, keywords = {Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}, number = {9}, publisher = {AIP Publishing}, title = {{Crises and chaotic scattering in hydrodynamic pilot-wave experiments}}, doi = {10.1063/5.0102904}, volume = {32}, year = {2022}, } @article{8999, abstract = {In many basic shear flows, such as pipe, Couette, and channel flow, turbulence does not arise from an instability of the laminar state, and both dynamical states co-exist. With decreasing flow speed (i.e., decreasing Reynolds number) the fraction of fluid in laminar motion increases while turbulence recedes and eventually the entire flow relaminarizes. The first step towards understanding the nature of this transition is to determine if the phase change is of either first or second order. In the former case, the turbulent fraction would drop discontinuously to zero as the Reynolds number decreases while in the latter the process would be continuous. For Couette flow, the flow between two parallel plates, earlier studies suggest a discontinuous scenario. In the present study we realize a Couette flow between two concentric cylinders which allows studies to be carried out in large aspect ratios and for extensive observation times. The presented measurements show that the transition in this circular Couette geometry is continuous suggesting that former studies were limited by finite size effects. A further characterization of this transition, in particular its relation to the directed percolation universality class, requires even larger system sizes than presently available. }, author = {Avila, Kerstin and Hof, Björn}, issn = {1099-4300}, journal = {Entropy}, number = {1}, publisher = {MDPI}, title = {{Second-order phase transition in counter-rotating taylor-couette flow experiment}}, doi = {10.3390/e23010058}, volume = {23}, year = {2021}, } @article{9407, abstract = {High impact epidemics constitute one of the largest threats humanity is facing in the 21st century. In the absence of pharmaceutical interventions, physical distancing together with testing, contact tracing and quarantining are crucial in slowing down epidemic dynamics. Yet, here we show that if testing capacities are limited, containment may fail dramatically because such combined countermeasures drastically change the rules of the epidemic transition: Instead of continuous, the response to countermeasures becomes discontinuous. Rather than following the conventional exponential growth, the outbreak that is initially strongly suppressed eventually accelerates and scales faster than exponential during an explosive growth period. As a consequence, containment measures either suffice to stop the outbreak at low total case numbers or fail catastrophically if marginally too weak, thus implying large uncertainties in reliably estimating overall epidemic dynamics, both during initial phases and during second wave scenarios.}, author = {Scarselli, Davide and Budanur, Nazmi B and Timme, Marc and Hof, Björn}, issn = {20411723}, journal = {Nature Communications}, number = {1}, publisher = {Springer Nature}, title = {{Discontinuous epidemic transition due to limited testing}}, doi = {10.1038/s41467-021-22725-9}, volume = {12}, year = {2021}, } @article{9558, abstract = {We show that turbulent dynamics that arise in simulations of the three-dimensional Navier--Stokes equations in a triply-periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than 10^5 degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system's observables as weighted sums over the periodic orbits. }, author = {Yalniz, Gökhan and Hof, Björn and Budanur, Nazmi B}, issn = {1079-7114}, journal = {Physical Review Letters}, number = {24}, publisher = {American Physical Society}, title = {{Coarse graining the state space of a turbulent flow using periodic orbits}}, doi = {10.1103/PhysRevLett.126.244502}, volume = {126}, year = {2021}, } @article{10299, abstract = {Turbulence generally arises in shear flows if velocities and hence, inertial forces are sufficiently large. In striking contrast, viscoelastic fluids can exhibit disordered motion even at vanishing inertia. Intermediate between these cases, a state of chaotic motion, “elastoinertial turbulence” (EIT), has been observed in a narrow Reynolds number interval. We here determine the origin of EIT in experiments and show that characteristic EIT structures can be detected across an unexpectedly wide range of parameters. Close to onset, a pattern of chevron-shaped streaks emerges in qualitative agreement with linear and weakly nonlinear theory. However, in experiments, the dynamics remain weakly chaotic, and the instability can be traced to far lower Reynolds numbers than permitted by theory. For increasing inertia, the flow undergoes a transformation to a wall mode composed of inclined near-wall streaks and shear layers. This mode persists to what is known as the “maximum drag reduction limit,” and overall EIT is found to dominate viscoelastic flows across more than three orders of magnitude in Reynolds number.}, author = {Choueiri, George H and Lopez Alonso, Jose M and Varshney, Atul and Sankar, Sarath and Hof, Björn}, issn = {1091-6490}, journal = {Proceedings of the National Academy of Sciences}, keywords = {multidisciplinary, elastoinertial turbulence, viscoelastic flows, elastic instability, drag reduction}, number = {45}, publisher = {National Academy of Sciences}, title = {{Experimental observation of the origin and structure of elastoinertial turbulence}}, doi = {10.1073/pnas.2102350118}, volume = {118}, year = {2021}, } @article{7932, abstract = {Pulsating flows through tubular geometries are laminar provided that velocities are moderate. This in particular is also believed to apply to cardiovascular flows where inertial forces are typically too low to sustain turbulence. On the other hand, flow instabilities and fluctuating shear stresses are held responsible for a variety of cardiovascular diseases. Here we report a nonlinear instability mechanism for pulsating pipe flow that gives rise to bursts of turbulence at low flow rates. Geometrical distortions of small, yet finite, amplitude are found to excite a state consisting of helical vortices during flow deceleration. The resulting flow pattern grows rapidly in magnitude, breaks down into turbulence, and eventually returns to laminar when the flow accelerates. This scenario causes shear stress fluctuations and flow reversal during each pulsation cycle. Such unsteady conditions can adversely affect blood vessels and have been shown to promote inflammation and dysfunction of the shear stress-sensitive endothelial cell layer.}, author = {Xu, Duo and Varshney, Atul and Ma, Xingyu and Song, Baofang and Riedl, Michael and Avila, Marc and Hof, Björn}, issn = {10916490}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, number = {21}, pages = {11233--11239}, publisher = {National Academy of Sciences}, title = {{Nonlinear hydrodynamic instability and turbulence in pulsatile flow}}, doi = {10.1073/pnas.1913716117}, volume = {117}, year = {2020}, } @article{8043, abstract = {With decreasing Reynolds number, Re, turbulence in channel flow becomes spatio-temporally intermittent and self-organises into solitary stripes oblique to the mean flow direction. We report here the existence of localised nonlinear travelling wave solutions of the Navier–Stokes equations possessing this obliqueness property. Such solutions are identified numerically using edge tracking coupled with arclength continuation. All solutions emerge in saddle-node bifurcations at values of Re lower than the non-localised solutions. Relative periodic orbit solutions bifurcating from branches of travelling waves have also been computed. A complete parametric study is performed, including their stability, the investigation of their large-scale flow, and the robustness to changes of the numerical domain.}, author = {Paranjape, Chaitanya S and Duguet, Yohann and Hof, Björn}, issn = {14697645}, journal = {Journal of Fluid Mechanics}, publisher = {Cambridge University Press}, title = {{Oblique stripe solutions of channel flow}}, doi = {10.1017/jfm.2020.322}, volume = {897}, year = {2020}, } @article{7534, abstract = {In the past two decades, our understanding of the transition to turbulence in shear flows with linearly stable laminar solutions has greatly improved. Regarding the susceptibility of the laminar flow, two concepts have been particularly useful: the edge states and the minimal seeds. In this nonlinear picture of the transition, the basin boundary of turbulence is set by the edge state's stable manifold and this manifold comes closest in energy to the laminar equilibrium at the minimal seed. We begin this paper by presenting numerical experiments in which three-dimensional perturbations are too energetic to trigger turbulence in pipe flow but they do lead to turbulence when their amplitude is reduced. We show that this seemingly counterintuitive observation is in fact consistent with the fully nonlinear description of the transition mediated by the edge state. In order to understand the physical mechanisms behind this process, we measure the turbulent kinetic energy production and dissipation rates as a function of the radial coordinate. Our main observation is that the transition to turbulence relies on the energy amplification away from the wall, as opposed to the turbulence itself, whose energy is predominantly produced near the wall. This observation is further supported by the similar analyses on the minimal seeds and the edge states. Furthermore, we show that the time evolution of production-over-dissipation curves provides a clear distinction between the different initial amplification stages of the transition to turbulence from the minimal seed.}, author = {Budanur, Nazmi B and Marensi, Elena and Willis, Ashley P. and Hof, Björn}, issn = {2469-990X}, journal = {Physical Review Fluids}, number = {2}, publisher = {American Physical Society}, title = {{Upper edge of chaos and the energetics of transition in pipe flow}}, doi = {10.1103/physrevfluids.5.023903}, volume = {5}, year = {2020}, }