Transition Threshold and the Self-Sustaining Process

Waleffe, Fabian; Wang, Jue

doi:10.1007/1-4020-4049-0_5

Cited by 16 publications

(13 citation statements)

References 25 publications

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“…These three elements are also the key ingredients of the MFFH interaction mechanism. Furthermore, numerical computations based on the Navier-Stokes equations for plane Couette flow at high Reynolds number (Waleffe & Wang 2005) find a lower branch of exact coherent structures that comprise streaks, streamwise rolls, and a 3D eigenmode, consistent with the MFFH theory; these coherent states are inherently nonlinear (do not bifurcate from the laminar flow) and provide an O(1/R) threshold for transition.…”

Section: Three-dimensional Effects Are Essential To Nonlinear Instabisupporting

confidence: 55%

David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

Akylas

2020

Annu. Rev. Fluid Mech.

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David J. Benney (1930–2015) was an applied mathematician and fluid dynamicist whose highly original work has shaped our understanding of nonlinear wave and instability processes in fluid flows. This article discusses the new paradigm he pioneered in the study of nonlinear phenomena, which transcends fluid mechanics, and it highlights the common threads of his research contributions, namely, resonant nonlinear wave interactions; the derivation of nonlinear evolution equations, including the celebrated nonlinear Schrödinger equation for modulated wave trains; and the significance of three-dimensional disturbances in shear flow instability and transition.

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Section: Three-dimensional Effects Are Essential To Nonlinear Instabisupporting

confidence: 55%

David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

Akylas

2020

Annu. Rev. Fluid Mech.

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“…Well above Re g it is expected that A c decreases as Re −γ when Re increases. A lower bound γ = 1 can be derived from a simple balance between linear and nonlinear terms in the Navier-Stokes equations but the actual value is highly debated, likely dependent on the flow geometry and a function of the by-pass scenario [114]; see [169] or the introduction of [49] for reviews.…”

Section: Nontrivial States On the Laminar-turbulent Boundarymentioning

confidence: 99%

On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular

Manneville

2015

European Journal of Mechanics - B/Fluids

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The main part of this contribution to the special issue of EJM-B/Fluids dedicated to Patrick Huerre outlines the problem of the subcritical transition to turbulence in wall-bounded flows in its historical perspective with emphasis on plane Couette flow, the flow generated between counter-translating parallel planes. Subcritical here means discontinuous and direct, with strong hysteresis. This is due to the existence of nontrivial flow regimes between the global stability threshold Re g , the upper bound for unconditional return to the base flow, and the linear instability threshold Re c characterized by unconditional departure from the base flow.The transitional range around Re g is first discussed from an empirical viewpoint ( §1). The recent determination of Re g for pipe flow by Avila et al. (2011) is recalled. Plane Couette flow is next examined. In laboratory conditions, its transitional range displays an oblique pattern made of alternately laminar and turbulent bands, up to a third threshold Re t beyond which turbulence is uniform.Our current theoretical understanding of the problem is next reviewed ( §2): linear theory and nonnormal amplification of perturbations; nonlinear approaches and dynamical systems, basin boundaries and chaotic transients in minimal flow units; spatiotemporal chaos in extended systems and the use of concepts from statistical physics, spatiotemporal intermittency and directed percolation, large deviations and extreme values. Two appendices present some recent personal results obtained in plane Couette flow about patterning from numerical simulations and modeling attempts.Keywords Transition to Turbulence; Pipe Flow; Plane Couette Flow; Laminar-Turbulent Patterning Cautionary note about the literature cited The number of articles related to the topics examined here is tremendous and, though already referring to more than 150 publications, the bibliography is far from exhaustive by at least one order of magnitude. For a better coverage, the reader is invited to consult the literature cited in the review papers mentioned. I tried not to bias the list according to my personal interests, while choosing what I thought to be the most representative papers in each subtopic, sometimes the most recent publications of given people or groups reviewing related works in their introductions and pointing backwards to earlier relevant papers. For convenience, references are listed in alphabetical order of the first author and next chronologically.The article published by O. Reynolds in 1883 [131] founded the scientific approach to the problem of the transition to turbulence. Already an abstract in itself, its title "An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and the law of resistance in parallel channels," summarized the main features of the problem and, between the words, identified its control parameter Re nowadays called the Reynolds number. This parameter is a measure of 1 arXiv:1403.6374v1 [physics.flu-dyn]

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“…The latter is typically of relative measure zero so orbits that lie on ∂B are rare but important since they lie just at the transition from the laminar flow toward something "more interesting." In particular, the threshold amplitude for transition, which has been a subject of some interest ( [10], [2], [1]), is the minimum distance from the origin to the basin boundary. We denote such a threshold point by T .…”

Section: B(x) Is Quadratic In X Andmentioning

confidence: 99%

Shear-flow transition: the basin boundary

Lebovitz

2009

Nonlinearity

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The basin of attraction of a stable equilibrium point is investigated for a dynamical system (W97) that has been used to model transition to turbulence in shear flows. The basin boundary contains a linearly unstable equilibrium point X lb which, in the self-sustaining scenario, plays a role in mediating the transition in that transition orbits cluster around its unstable manifold. We find for W97, however, that this role is played not by X lb but rather by a periodic orbit also lying on the basin boundary. Moreover, it appears via numerical computations that all orbits beginning near X lb relaminarize. We offer evidence that this is due to the exquisite narrowness of the complementary region to the basin of attraction in the part of phase space near X lb . This further leads to a proposal for understanding the 'edge of chaos' in terms of more familiar invariant sets of a dynamical system.

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Transition Threshold and the Self-Sustaining Process

Cited by 16 publications

References 25 publications

David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular

Shear-flow transition: the basin boundary

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