Towards a Two-Fluid Picture of Intermittency in Shell Models of Turbulence

Gilson, Jean-Louis; Dombre, T.

doi:10.1103/physrevlett.79.5002

Cited by 14 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The blowup in the Sabra shell model has self-similar universal structure [16]. Possible relation of such structure to the statistics of turbulence was discussed in [17,18]. Cascade models of turbulence [19] represent the extended version of shell models, where each shell is described by a large (though fixed) number of variables.…”

Section: Introductionmentioning

confidence: 99%

Blowup as a driving mechanism of turbulence in shell models

Mailybaev

2013

Phys. Rev. E

View full text Add to dashboard Cite

Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of the mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949, Onsager noticed that the scaling properties in the inertial range imply nondifferentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup indeed represents the driving mechanism of the inertial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inertial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations.

show abstract

Section: Introductionmentioning

confidence: 99%

Blowup as a driving mechanism of turbulence in shell models

Mailybaev

2013

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…The possible role of structures as "atoms" for intermittency has recently been addressed in Ref. [14] for shell models (for white-in-time shell models see Ref. [15]).…”

mentioning

confidence: 99%

Structures and Intermittency in a Passive Scalar Model

Vergassola

Mazzino

1997

Phys. Rev. Lett.

View full text Add to dashboard Cite

A one-dimensional white-in-time passive scalar model is introduced. Strong and persistent structures are shown to be present. A perturbative expansion for the scaling exponents is performed around a Gaussian limit of the model. The resulting predictions are compared with numerical simulations. PACS number(s) : 47.10.+g, 47.27.-i, 05.40.+j It is commonly observed in turbulent flows that coherent and well localized structures emerge in the sea of disorganized, random regions. Well-known examples are provided by 2D and 3D Navier-Stokes turbulence : numerical simulations show that structures appear respectively in the form of quasi-circular vortices and thin, elongated vortex tubes. No ab initio theory for the statistical properties of the flows (and a fortiori of their dependence on the structures) is however available. A systematic statistical theory has recently been developed for white-in-time passive field models. Their crucial property is that simultaneous correlation functions obey closed equations of motion [1]. Anomalous scaling and intermittency have in particular been considered. The general mechanism for the appearance of anomalous scaling is associated with zero modes of the equations for the correlation functions [2,3,4]. For Kraichnan passive scalar model, anomalies first appear at the level of fourth-order correlations [1 -3, 5]. When a large-scale gradient is imposed, also the third-order moment is anomalous [6]. Scaling exponents have been calculated by using perturbative expansions around Gaussian limits [2,3]. For magnetic fields a nonperturbative anomalous solution for second-order correlations has been found [7]. The zero mode responsible for anomalous scaling comes from the balance in the inertial range between field lines stretching and eddy-damping. This suggests that also for white-in-time models, structures could be present and play an important role. A crucial 1

show abstract

“…It could also be interesting to investigate the relation between coherent structures in the GOY model [21,20] and those in the passive scalar model.…”

Section: Discussionmentioning

confidence: 99%

“…where D θ nm is the dissipation term, very similar to the one before D θ nm = Dk 2 n δ nm (20) and B nm is the contribution of the cascade term given by:…”

Section: Scaling Of Eigenvalue Spectramentioning

confidence: 99%

Fixed points, stability, and intermittency in a shell model for advection of passive scalars

Kockelkoren¹,

Jensen²

2000

Phys. Rev. E

View full text Add to dashboard Cite

We investigate the fixed points of a shell model for the turbulent advection of passive scalars introduced in Jensen, Paladin, and Vulpiani [Phys. Rev. A 45, 7214 (1992)]. The passive scalar field is driven by the velocity field of the popular Gledzer-Ohkitani-Yamada (GOY) shell model. The scaling behavior of the static solutions is found to differ significantly from Obukhov-Corrsin scaling straight theta(n) approximately k(-1/3)(n), which is only recovered in the limit where the diffusivity vanishes, D-->0. From the eigenvalue spectrum we show that any perturbation in the scalar will always damp out, i.e., the eigenvalues of the scalar are negative and are decoupled from the eigenvalues of the velocity. We estimate Lyapunov exponents and the intermittency parameters using a definition proposed by Benzi, Paladin, Parisi, and Vulpiani [J. Phys. A 18, 2157 (1985)]. The full model is found to be as chaotic as the GOY model, measured by the maximal Lyapunov exponent, but is more intermittent.

show abstract

Towards a Two-Fluid Picture of Intermittency in Shell Models of Turbulence

Cited by 14 publications

References 12 publications

Blowup as a driving mechanism of turbulence in shell models

Blowup as a driving mechanism of turbulence in shell models

Structures and Intermittency in a Passive Scalar Model

Fixed points, stability, and intermittency in a shell model for advection of passive scalars

Contact Info

Product

Resources

About