Physical model of intermittency in turbulence: Near-dissipation-range non-Gaussian statistics

Abstract

We present a physical model constructed from the Navier-Stokes equation to describe the evolution of the probability distribution function of transverse velocity gradients in 3D isotropic turbulence. Quantitative agreement with data from direct numerical simulations of isotropic turbulence for a wide range of Reynolds number is obtained. The model is based on a concrete physical picture of self-distortion of structures and interaction between random eddies and structures; the dynamical balance explains the non-Gaussian equilibrium probability distributions. PACS numbers: 47.25.Cg, 02.50.+s, 05.40.+j The non-Gaussian statistical nature of small scales in turbulence has been a major obstacle in building a wellfounded statistical theory. The probability distribution functions (PDFs) for the spatial velocity gradients or velocity diA'erences across a small distance, measured for Bows in both laboratories and numerical simulations, show considerably longer "tails" than the bell curve of a Gaussian distribution.For quite a long time, there has been a lack of any quantitative description of the observed non-Gaussian PDFs. Recently, several theoretical approaches have been developed to address this question. In a newly developed mapping-closure theory, Kraichnan introduced the idea of keeping track of the development of non-Gaussian statistics by using a Gaussian reference field and a mapping function. The dynamical evolution of the PDF is described by an evolution equation of the mapping function; the latter is obtained directly from the original dynamical equation under certain statistical hypotheses. This theory gives a good quantitative description of the non-Gaussian PDF of the velocity gradient under Burgers dynamics.In this Letter, we show that Kraichnan's idea can be fruitfully carried over to quantitatively describe the observed nonGaussian PDF of the transverse velocity gradients in 3DNavier-Stokes turbulence. The present model equation for a mapping function is obtained by plausible physical assumptions about the basic dynamical processes behind the Navier-Stokes equation. In particular, we explicitly address the importance of local-versus-nonlocal interaction in physical space in developed turbulence. Strikingly, the simple model gives a quantitatively good description of PDFs observed in direct numerical simulations of isotropic turbulence over a wide range of Reynolds numbers available within the current power of supercomputers.Our canonical picture is as follows:Fully developed turbulence consists of both hierarchies of random eddies and small-scale coherent structures. Hereafter, we refer to them simply as eddies and structures, respectively. Eddies are statistically (almost) independent, and thus have near-Gaussian statistics. Important dynamical properties of eddies, like the energy transfer, are describable by near-Gaussian statistical closure models such as the direct-interaction and renormalization-group approximations. On the other hand, structures, appearing locally and intermittently in space a...

Fundamental statistical descriptions of plasma turbulence in magnetic fields

“…A heuristic model of the PDF of the transverse velocity gradient for Navier-Stokes turbulence based on these ideas was very encouraging (Kraichnan, 1990). That work was pursued by She (1991a) and She and Orszag (1991); for a summary and further discussion, see She (1991b). In plasma physics the first published work on mapping closures is due to Das and Kaw (1995).…”

Probability Density Functions of Velocity Increments in the Atmospheric Boundary Layer

“…Various theoretical interpretations have been proposed. Based on a physical picture of fully developed turbulence as a collection of weakly correlated random eddies and strongly correlated structured eddies, She (1991) (see also, She and Orszag 1991) proposed a model to describe the evolution of the pdf's of transverse velocity gradients in three-dimensional isotropic turbulence. From an analytical closure for Burgers turbulence, Kraichnan (1990) also proposed a model for the evolution of the pdf's of velocity increments.…”

Emergence of coherent patterns of vortex stretching during reconnection: A scattering paradigm

“…Recently, a discussion of local vortex self-stretching, intermittency, and their relation to non-Gaussian statistics of small scales has been given by She and co-workers [17][18][19]. Our high-resolution simulations suggest that the self-stretching regions are part of the reconnection process.…”