Transition to turbulence when the Tollmien–Schlichting and bypass routes coexist

Zammert, Stefan; Eckhardt, Bruno

doi:10.1017/jfm.2019.724

Cited by 8 publications

(17 citation statements)

References 75 publications

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“…In particular, no matter how small or how large the bound a L , provided the computational box is long enough there will always be another trajectory with a < a L also experiencing transition, possibly featuring destabilisation of TS waves. A similar situation was described by Zammert & Eckhardt (2017b) the type of transition was determined from the knowledge of pre-defined transition times.…”

Section: Discussionmentioning

confidence: 97%

“…The edge trajectory is by construction linearly unstable. So is the laminar base flow, except that the instabilities manifest themselves over different timescales, with the base flow instability being typically slower (Zammert & Eckhardt 2017b). If the time horizon in the bisection is large enough there is a competition between bypass transition (expected from the linear instability of the edge) and the classical transition scenario (featuring Tollmien-Schlichting waves, due to the linear instability of the base flow).…”

Section: Physical Spacementioning

confidence: 99%

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Edge tracking in spatially developing boundary layer flows

et al. 2019

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Recent progress in understanding subcritical transition to turbulence is based on the concept of the edge, the manifold separating the basins of attraction of the laminar and the turbulent state. Originally developed in numerical studies of parallel shear flows with a linearly stable base flow, this concept is adapted here to the case of a spatially developing Blasius boundary layer. Longer time horizons fundamentally change the nature of the problem due to the loss of stability of the base flow due to Tollmien-Schlichting (TS) waves. We demonstrate, using a moving box technique, that efficient long-time tracking of edge trajectories is possible for the parameter range relevant to bypass transition, even if the asymptotic state itself remains out of reach. The flow along the edge trajectory features streak switching observed for the first time in the Blasius boundary layer. At long enough times, TS waves co-exist with the coherent structure characteristic of edge trajectories. In this situation we suggest a reinterpretation of the edge as a manifold dividing the state space between the two main types of boundary layer transition, i.e. bypass transition and classical transition.

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Section: Discussionmentioning

confidence: 97%

Section: Physical Spacementioning

confidence: 99%

Edge tracking in spatially developing boundary layer flows

et al. 2019

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“…The local edge tracking algorithms do not rely on a Boolean approach, but can be defined as optimizers of a scalar observable. We point out that the identification of local edge properties is not limited to bistable systems, and therefore as future work a revisit of manifold-tracking methods in different situations such as [24,31,73] using this framework is strongly encouraged.…”

Section: Conclusion Discussion and Outlookmentioning

confidence: 99%

“…Global methods can, however, fall short whenever no available global observable can elucidate on which side of the edge the different trajectories evolve [31,73], when one of the attractors undergoes a local bifurcation affecting the values of the bounds [74] or simply when information on the bounds α A,B is not available.…”

Section: A Global Versus Local Methodsmentioning

confidence: 99%

Edge manifold as a Lagrangian coherent structure in a high-dimensional state space

Beneitez

Duguet

Schlatter

et al. 2020

Phys. Rev. Research

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Dissipative dynamical systems characterized by two basins of attraction are found in many physical systems, notably in hydrodynamics where laminar and turbulent regimes can coexist. The state space of such systems is structured around a dividing manifold called the edge, which separates trajectories attracted by the laminar state from those reaching the turbulent state. We apply here concepts and tools from Lagrangian data analysis to investigate this edge manifold. This approach is carried out in the state space of autonomous arbitrarily highdimensional dissipative systems, in which the edge manifold is reinterpreted as a Lagrangian coherent structure (LCS). Two different diagnostics, finite-time Lyapunov exponents and Lagrangian descriptors, are used and compared with respect to their ability to identify the edge and their scalability. Their properties are illustrated on several low-order models of subcritical transition of increasing dimension and complexity, as well on wellresolved simulations of the Navier-Stokes equations in the case of plane Couette flow. They allow for a mapping of the global structure of both the state space and the edge manifold based on quantitative information. Both diagnostics can also be used to generate efficient bisection algorithms to approach asymptotic edge states, which outperform classical edge tracking.

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“…The edge manifold generalises hence to the separatrix between the turbulent state and a new attractor replacing dynamically the traditional laminar fixed point. In plane Poiseuille flow, for the parameters chosen in the corresponding study, the edge state is a travelling wave solution appearing in a saddle-node bifurcation at a finite Reynolds number R ≈ 459, while the instability of the base flow to TS waves does not occur before R = 5815 [26]. As R is increased from low to high values, the classical route emerges in parallel to the bypass route, and concerns an increasingly large subset of the state space of initial conditions [26].…”

Section: Introductionmentioning

confidence: 99%

A model for the collapse of the edge when two transitions routes compete

Beneitez,

Duguet,

Henningson

2020

Preprint

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The transition to turbulence in many shear flows proceeds along two competing routes, one linked with finite-amplitude disturbances and the other one originating from a linear instability, as in e.g. boundary layer flows. The dynamical systems concept of edge manifold has been suggested in the subcritical case to explain the partition of the state space of the system. This investigation is devoted to the evolution of the edge manifold when a linear stability is added in such subcritical systems, a situation poorly studied despite its prevalence in realistic fluid flows. In particular the fate of the edge state as a mediator of transition is unclear. A deterministic three-dimensional model is suggested, parametrised by the linear instability growth rate. The edge manifold evolves topologically, via a global saddle-loop bifurcation, from the separatrix between two attraction basins to the mediator between two transition routes. For larger instability rates, the stable manifold of the saddle point increases in codimension from 1 to 2 after an additional local saddle node bifurcation, causing the collapse of the edge manifold. As the growth rate is increased, three different regimes of this model are identified, each one associated with a flow case from the recent hydrodynamic literature. A simple nonautonomous generalisation of the model is also suggested in order to capture the complexity of spatially developing flows.

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Transition to turbulence when the Tollmien–Schlichting and bypass routes coexist

Cited by 8 publications

References 75 publications

Edge tracking in spatially developing boundary layer flows

Edge tracking in spatially developing boundary layer flows

Edge manifold as a Lagrangian coherent structure in a high-dimensional state space

A model for the collapse of the edge when two transitions routes compete

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