Renormalization and universality of blowup in hydrodynamic flows

Mailybaev, Alexei A.

doi:10.1103/physreve.85.066317

Cited by 17 publications

(33 citation statements)

References 32 publications

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“…This statement was recently supported in direct numerical simulation of the NSE with appropriate dynamical mode reduction [24][25][26][27] and in the equivalent helical version of shell models [28]. Let us note that instantonic solutions were shown to be closely related to the events preceding a shock formation in compressible flows [29], justifying their relevance also for realistic hydrodynamical systems in the continuum. Such a relation for incompressible flows, as well as for the case of a chaotic instanton, is unknown.…”

Section: Introductionmentioning

confidence: 69%

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Chaotic and regular instantons in helical shell models of turbulence

2017

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Shell models of turbulence have a finite-time blowup in the inviscid limit, i.e., the enstrophy diverges while the single-shell velocities stay finite. The signature of this blowup is represented by self-similar instantonic structures traveling coherently through the inertial range. These solutions might influence the energy transfer and the anomalous scaling properties empirically observed for the forced and viscous models. In this paper we present a study of the instantonic solutions for a set of four shell models of turbulence based on the exact decomposition of the Navier-Stokes equations in helical eigenstates. We find that depending on the helical structure of each model, instantons are chaotic or regular. Some instantonic solutions tend to recover mirror symmetry for scales small enough. Models that have anomalous scaling develop regular non chaotic instantons. Conversely, models that have non anomalous scaling in the stationary regime are those that have chaotic instantons. The direction of the energy carried by each single instanton tends to coincide with the direction of the energy cascade in the stationary regime. Finally, we find that whenever the small-scale stationary statistics is intermittent, the instanton is less steep than the dimensional Kolmogorov scaling, independently of whether or not it is chaotic. Our findings further support the idea that instantons might be crucial to describe some aspects of the multi-scale anomalous statistics of shell models

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Section: Introductionmentioning

confidence: 69%

“…In this paper we are interested in understanding the propagation of fluctuations in the inertial range of scales, i.e., in the inviscid limit. In such a limit, solutions of shell models are characterized by a finite-time infinite growth (blowup) of the enstrophy [13,29,32,33]:…”

Section: Finite-time Blowup In the Inviscid Modelmentioning

confidence: 99%

Chaotic and regular instantons in helical shell models of turbulence

2017

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“…One can use a symmetry group of the Burgers equation, which includes shifts of origin, scale changes and the Galilean transformation, to simplify the blowup description. In generic case, this reduces the initial condition to the form [39,32]. Substituting this expression into Eq.…”

Section: Internal "Clock" Of the Blowupmentioning

confidence: 99%

Spontaneously stochastic solutions in one-dimensional inviscid systems

Mailybaev

2016

Nonlinearity

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In this paper, we study the inviscid limit of the Sabra shell model of turbulence, which is considered as a particular case of a viscous conservation law in one space dimension with a nonlocal quadratic flux function. We present a theoretical argument (with a detailed numerical confirmation) showing that a classical deterministic solution before a finitetime blowup, t < t b , must be continued as a stochastic process after the blowup, t > t b , representing a unique physically relevant description in the inviscid limit. This theory is based on the dynamical system formulation written for the logarithmic time τ = log(t−t b ), which features a stable traveling wave solution for the inviscid Burgers equation, but a stochastic traveling wave for the Sabra model. The latter describes a universal onset of stochasticity immediately after the blowup.

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“…Blowup remains an active area of numerical research [19][20][21], but computational limitations are still the major obstacle. See also [22,23] for the blowup at a physical boundary, which is a related but different problem.Numerical limitations of the DNS can be overcome using simplified models [24][25][26], which were developed in lower spatial dimensions [27,28] or by exploring the cascade ideas in so-called shell models [29][30][31]. The reduced wave vector set approximation (REWA) model introduced in [32,33] restricted the Euler or Navier-Stokes dynamics to a self-similar set of wave vectors.…”

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confidence: 99%

Chaotic Blowup in the 3D Incompressible Euler Equations on a Logarithmic Lattice

Campolina

Mailybaev

2018

Phys. Rev. Lett.

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The dispute on whether the three-dimensional (3D) incompressible Euler equations develop an infinitely large vorticity in a finite time (blowup) keeps increasing due to ambiguous results from state-of-the-art direct numerical simulations (DNS), while the available simplified models fail to explain the intrinsic complexity and variety of observed structures. Here, we propose a new model formally identical to the Euler equations, by imitating the calculus on a 3D logarithmic lattice. This model clarifies the present controversy at the scales of existing DNS and provides the unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor spans over the anomalously large six-decade interval of spatial scales. For the original Euler system, our results suggest that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) are unsuitable, by far, for the blowup analysis, and establish new fundamental requirements for the approach to this long-standing problem.

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Renormalization and universality of blowup in hydrodynamic flows

Cited by 17 publications

References 32 publications

Chaotic and regular instantons in helical shell models of turbulence

Chaotic and regular instantons in helical shell models of turbulence

Spontaneously stochastic solutions in one-dimensional inviscid systems

Chaotic Blowup in the 3D Incompressible Euler Equations on a Logarithmic Lattice

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