Transition to turbulence in shear flows

Eckhardt, Bruno; Marzinzik, K.; Schmiegel, Armin

doi:10.1007/bfb0104973

Cited by 13 publications

(15 citation statements)

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“…While eventual decay is observed in most cases, the fluid response is chaotic in the interim. They further show that simulations with the same total initial energy and Re numbers but with differing initial conditions can exhibit vastly differing decay timescales -and in some instances there is no dissipation over the duration of a simulation (Eckhardt et al 1998). Though not reported here in detail, we also observe similar sensitivity to initial conditions for simulations with finite Re numbers in the way described in their work.…”

Section: Tg: Its Linear Decay and Its Nonlinear Reccurrencesupporting

confidence: 75%

“…4. There exist numerical simulations of plane Couette flows, which have revealed the details of what can be called "subcritical turbulence" in these flows (Schmiegel & Eckhardt 1997;Eckhardt et al 1998). These calculations, which are the only ones (as far as we know) of this kind and extend for reasonably long integration times, indicate that above a critical Reynolds number there appears dynamical activity, resulting probably from repeated TG events, brought about by nonlinear interaction and persisting much longer than the viscous decay time, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results

Umurhan¹,

Regev²

2004

A&A

123

193

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Abstract. We present here both analytical and numerical results of hydrodynamic stability investigations of rotationally supported circumstellar flows using the shearing box formalism. Asymptotic scaling arguments justifying the shearing box approximation are systematically derived, showing that there exist two limits which we call small shearing box (SSB) and large shearing box (LSB). The physical meaning of these two limits and their relationship to model equations implemented by previous investigators are discussed briefly. Two dimensional (2D) dynamics of the SSB are explored and shown to contain transiently growing (TG) linear modes, whose nature is first discussed within the context of linear theory. The fully nonlinear regime in 2D is investigated numerically for very high Reynolds (Re) numbers. Solutions exhibiting long-term dynamical activity are found and manifest episodic but recurrent TG behavior and these are associated with the formation and long-term survival of coherent vortices. The life-time of this spatio-temporal complexity depends on the Re number and the strength and nature of the initial disturbance. The dynamical activity in finite Re solutions ultimately decays with a characteristic time increasing with Re. However, for large enough Re and appropriate initial perturbation, a large number of TG episodes recur before any viscous decay begins to clearly manifest itself. In cases where Re = ∞ nominally (i.e. any dissipation resulting only from numerical truncation errors), the dynamical activity persists for the entire duration of the simulation (hundreds of box orbits). Because the SSB approximation used here is equivalent to a 2D incompressible flow, the dynamics can not depend on the Coriolis force. Therefore, three dimensional (3D) simulations are needed in order to decide if this force indeed suppresses nonlinear hydrodynamical instability in rotationally supported disks in the shearing box approximation, and if recurrent TG behavior can still persist in three dimensions as well -possibly giving rise to a subcritical transition to long-term spatio-temporal complexity.

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Section: Tg: Its Linear Decay and Its Nonlinear Reccurrencesupporting

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results

Umurhan¹,

Regev²

2004

A&A

123

193

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“…Besides, similarly to the MRI dynamo, the typical lifetime of turbulent dynamics in Poiseuille flow is finite, but increases exponentially with Re (Hof et al 2006). Perhaps not surprisingly considering its dynamical complexity, this nonlinear hydrodynamic transition remained very poorly understood for more than a century despite its relevance to many applied fluid dynamics problems, and it was not until the early 1990s that a consistent phenomenological picture started to emerge (see reviews by Eckhardt et al 2007;Eckhardt 2009;Mullin 2011;Eckhardt 2018). An important milestone on this problem, which as we are about to discover is also strongly relevant to instability-driven dynamos, was the numerical discovery and subsequent characterisation of a three-stage dynamical regeneration cycle, now commonly referred to as the self-sustaining process (SSP, see Hamilton, Kim & Waleffe 1995;Waleffe 1995Waleffe , 1997.…”

Section: Self-sustaining Nonlinear Processesmentioning

confidence: 99%

Dynamo theories

2019

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These lecture notes are based on a tutorial given in 2017 at a plasma physics winter school in Les Houches. Their aim is to provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field. The primary focus is on the physical and mathematical concepts underlying different (turbulent) branches of dynamo theory, with some astrophysical, geophysical and experimental contexts disseminated throughout the document. The text begins with an introduction to the rationale, observational and historical roots of the subject, and to the basic concepts of magnetohydrodynamics relevant to dynamo theory. The next two sections discuss the fundamental phenomenological and mathematical aspects of (linear and nonlinear) small- and large-scale magnetohydrodynamic (MHD) dynamos. These sections are complemented by an overview of a selection of current active research topics in the field, including the numerical modelling of the geo- and solar dynamos, shear dynamos driven by turbulence with zero net helicity and MHD-instability-driven dynamos such as the magnetorotational dynamo. The difficult problem of a unified, self-consistent statistical treatment of small- and large-scale dynamos at large magnetic Reynolds numbers is also discussed throughout the text. Finally, an excursion is made into the relatively new but increasingly popular realm of magnetic-field generation in weakly collisional plasmas. A short discussion of the outlook and challenges for the future of the field concludes the presentation.

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“…Fortunately, a family of such flows exists, in the recently-discovered "exact coherent states" (ECS) found by computational bifurcation analysis in plane Couette and plane Poiseuille flows [12,13,14,15,16]. These are three-dimensional, traveling wave flows (hence steady in a traveling reference frame) that appear via saddle-node bifurcations [35] at a Reynolds number somewhat below the transition value seen in experiments [17,18].…”

mentioning

confidence: 99%

Toward a Structural Understanding of Turbulent Drag Reduction: Nonlinear Coherent States in Viscoelastic Shear Flows

Stone

Waleffe²,

Graham

2002

Phys. Rev. Lett.

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Nontrivial steady flows have recently been found that capture the main structures of the turbulent buffer layer. We study the effects of polymer addition on these "exact coherent states" (ECS) in plane Couette flow. Despite the simplicity of the ECS flows, these effects closely mirror those observed experimentally: structures shift to larger length scales, wall-normal fluctuations are suppressed while streamwise ones are enhanced, and drag is reduced. The mechanism underlying these effects is elucidated. These results suggest that the ECS are closely related to buffer layer turbulence.PACS numbers: 83.60. Yz,83.80.Rs,47.20.Ky,47.27.Cn Rheological drag reduction, the suppression by additives of skin friction in turbulent flow, has received much attention since its discovery in 1947 [1,2,3]. For many polymer-solvent systems, the pressure drop measured in the pipe flow of the solution can be 30 − 50% less than for the solvent alone. The central rheological feature of drag-reducing additives is their extensional behavior in solution: for dilute polymer solutions in particular the stresses arising in extensional flow can be orders of magnitude larger than those developed in a shear flow. This fact is well-recognized; nevertheless the mechanism of interaction between polymer stretching and turbulent structure is not well-understood and the goal of the present work is to attempt to shed light on this interaction.A key structural observation from experiments and direct numerical simulations (DNS) of drag-reducing solutions is the modification of the buffer region near the wall [4,5,6,7,8,9,10]. It has long been known that the flow in this region is very structured, containing streamwise vortices that lead to streaks in the streamwise velocity [11]; these structures are thickened in both the wall-normal and spanwise directions during flow of drag reducing solutions [4,5]. Because of its importance in the production and dissipation of turbulent energy [11], any effort to mechanistically understand rheological drag reduction should address this region.To better understand the effect of the polymer on the buffer layer, we wish to study a model flow that has structures similar to those seen in this region but without the full complexities of time-dependent turbulent flows. Fortunately, a family of such flows exists, in the recently-discovered "exact coherent states" (ECS) found by computational bifurcation analysis in plane Couette and plane Poiseuille flows [12,13,14,15,16]. These are three-dimensional, traveling wave flows (hence steady in a traveling reference frame) that appear via saddle-node bifurcations [35] at a Reynolds number somewhat below the transition value seen in experiments [17,18]. The structure of the ECS captures the counter-rotating staggered streamwise vortices that dominate the structure in the buffer region. From the dynamical point of view, there is evidence that these states form a part of the dynamical skeleton of the turbulent flow: i.e., they are saddle points that underlie the strange attract...

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Transition to turbulence in shear flows

Cited by 13 publications

References 43 publications

Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results

Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results

Dynamo theories

Toward a Structural Understanding of Turbulent Drag Reduction: Nonlinear Coherent States in Viscoelastic Shear Flows

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