Stochastic Navier–Stokes Equation with Colored Noise: Renormalization Group Analysis

N V Antonov,A V Malyshev,N M Gulitskiy,L Bravina,Y Foka,S Kabana

doi:10.1051/epjconf/201612604019

Abstract

In this work we study the fully developed turbulence described by the stochastic Navier–Stokes equation with finite correlation time of random force. Inertial-range asymptotic behavior is studied in one-loop approximation and by means of the field theoretic renormalization group. The inertial-range behavior of the model is described by limiting case of vanishing correlation time that corresponds to the nontrivial fixed point of the RG equation. Another fixed point is a saddle type point, i.e., it is infrared attractive only in one of two possible directions. The existence and stability of fixed points depends on the relation between the exponents in the energy spectrum e ∝ k 1− y and the dispersion law ω ∝ k 2− η .

Highlights

Introduction and description of the modelOne of the possible ways to consider the fully developed turbulence within the framework of some microscopic model is to study the stochastic Navier–Stokes equation with random external force [1]
From (3) it follows that the energy spectrum of the velocity in the inertial range has the form E ∝ k1−y, while the correlation time at the momentum k scales as k−2+η
In a number of papers this approach was applied to the case of passive vector fields advected by a turbulent flow with some prescribed properties: large-scale anisotropy, helicity, compressibility, finite correlation time, non-Gaussianity, a more general form of nonlinearity, etc.; see [12,13,14,15,16] and references therein

Summary

Introduction and description of the model

One of the possible ways to consider the fully developed turbulence within the framework of some microscopic model is to study the stochastic Navier–Stokes equation with random external force [1]. From (3) it follows that the energy spectrum of the velocity in the inertial range has the form E ∝ k1−y, while the correlation time at the momentum k scales as k−2+η This means that the random force φi is simulated by the following statistical ensemble: it is assumed to be Gaussian, homogeneous, with the zero mean and the following correlation function: φi(t, x)φ j(t , x ) = D0 dω 2π k>m dk (2π)d k8−d−(y+2η) Pi j(k) ω2 + ν20u20k4−2η eik·(x−x )−iω(t−t ). Define the coupling constant g0, which plays the role of the expansion parameter in the ordinary perturbation theory, and the characteristic ultraviolet (UV) momentum scale Λ Such ensemble with arbitrary function Dφ(ω, k) ∝ f Wk2/ω3 instead of (3) was used in [9] (here W is dissipation rate of energy). In a number of papers this approach was applied to the case of passive vector (magnetic) fields advected by a turbulent flow with some prescribed properties: large-scale anisotropy, helicity, compressibility, finite correlation time, non-Gaussianity, a more general form of nonlinearity, etc.; see [12,13,14,15,16] and references therein

Field theoretic formulation of the model

Feynman diagrammatic technique

Renormalization of the model and fixed points

Critical dimensions

Conclusion