Probability Densities in Strong Turbulence

According to modern developments in turbulence theory, the "dissipation" scales (u.v. cut-offs) $\eta$ form a random field related to velocity increments $\delta_{\eta}u$. In this work we, using Mellin's transform combined with the Gaussain large -scale boundary condition, calculate probability densities (PDFs) of velocity increments $P(\delta_{r}u,r)$ and the PDF of the dissipation scales $Q(\eta, Re)$, where $Re$ is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF $P_{L}(\delta_{r}u,r)$ often quoted in the literature. It is shown that the probability density of the small-scale velocity fluctuations includes information about the large (integral) scale dynamics which is responsible for deviation of $P(\delta_{r}u,r)$ from $P_{L}(\delta_{r}u,r)$. A framework for evaluation of the PDFs of various turbulence characteristics involving spatial derivatives is developed. The exact relation, free of spurious Logarithms recently discussed in Frisch et al (J. Fluid Mech. {\bf 542}, 97 (2005)), for the multifractal probability density of velocity increments, not based on the steepest descent evaluation of the integrals is obtained and the calculated function $D(h)$ is close to experimental data. A novel derivation (Polyakov, 2005), of a well-known result of the multi-fractal theory [Frisch, "Turbulence. {\it Legacy of A.N.Kolmogorov}", Cambridge University Press, 1995)), based on the concepts described in this paper, is also presented.Comment: 25 pages and 9 figure

Development of turbulence models for shear flows by a double expansion technique

Turbulence models are developed by supplementing the renormalization group (RNG) approach of Yakhot and Orszag with scale expansions for the Reynolds stress and production of dissipation terms. The additional expansion parameter (eta) is the ratio of the turbulent to mean strain time scale. While low-order expansions appear to provide an adequate description of the Reynolds stress, no finite truncation of the expansion for the production of dissipation term in powers of eta suffices - terms of all orders must be retained. Based on these ideas, a new two-equation model and Reynolds stress transport model are developed for turbulent shear flows. The models are tested for homogeneous shear flow and flow over a backward facing step. Comparisons between the model predictions and experimental data are excellent

Mean- Field Approximation and a Small Parameter in Turbulence Theory

Numerical and physical experiments on two-dimensional (2d) turbulence show that the differences of transverse components of velocity field are well described by a gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume existence of critical space-dimensionality $d=d_{c}$, at which the energy flux and all odd-order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant. At $d<d_{c}$ the flow can be described by the ``mean-field theory'', leading to the observed gaussian statistics and Kolmogorov scaling of transverse velocity differences. It is shown that in the vicinity of $d=d_{c}$ the ratio of the relaxation and translation characteristic times decreases to zero, thus giving rise to a small parameter of the theory. The expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of $d=d_{c}$. The resulting equation describes experimental data on two-dimensional turbulence and demonstrate onset of intermittency as $d-d_{c}>0$ and $r/L\to 0$ in three-dimensional flows in close agreement with experimental data. In addition, some new exact relations between correlation functions of velocity differences are derived. It is also predicted that the single-point pdf of transverse velocity difference in developing as well as in the large-scale stabilized two-dimensional turbulence is a gaussian.Comment: 25 pages, 1 figur

Large eddy simulation of two-dimensional isotropic turbulence

Large eddy simulation (LES) of forced, homogeneous, isotropic, two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlov et al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary djnlx.tex fil

Large-Eddy Simulations of Fluid and Magnetohydrodynamic Turbulence Using Renormalized Parameters

In this paper a procedure for large-eddy simulation (LES) has been devised for fluid and magnetohydrodynamic turbulence in Fourier space using the renormalized parameters. The parameters calculated using field theory have been taken from recent papers by Verma [Phys. Rev. E, 2001; Phys. Plasmas, 2001]. We have carried out LES on $64^3$ grid. These results match quite well with direct numerical simulations of $128^3$. We show that proper choice of parameter is necessary in LES.Comment: 12 pages, 4 figures: Proper figures inserte

Local shell-to-shell energy transfer via nonlocal Interactions in fluid turbulence

In this paper we analytically compute the strength of nonlinear interactions in a triad, and the energy exchanges between wavenumber shells in incompressible fluid turbulence. The computation has been done using first-order perturbative field theory. In three dimension, magnitude of triad interactions is large for nonlocal triads, and small for local triads. However, the shell-to-shell energy transfer rate is found to be local and forward. This result is due to the fact that the nonlocal triads occupy much less Fourier space volume than the local ones. The analytical results on three-dimensional shell-to-shell energy transfer match with their numerical counterparts. In two-dimensional turbulence, the energy transfer rates to the near-by shells are forward, but to the distant shells are backward; the cumulative effect is an inverse cascade of energy.Comment: 10 pages, Revtex

Anisotropy in Turbulent Flows and in Turbulent Transport

We discuss the problem of anisotropy and intermittency in statistical theory of high Reynolds-number turbulence (and turbulent transport). We present a detailed description of the new tools that allow effective data analysis and systematic theoretical studies such as to separate isotropic from anisotropic aspects of turbulent statistical fluctuations. Employing the invariance of the equations of fluid mechanics to all rotations, we show how to decompose the (tensorial) statistical objects in terms of the irreducible representation of the SO(3) symmetry group. For the case of turbulent advection of passive scalar or vector fields, this decomposition allows rigorous statements to be made: (i) the scaling exponents are universal, (ii) the isotropic scaling exponents are always leading, (iii) the anisotropic scaling exponents form a discrete spectrum which is strictly increasing as a function of the anisotropic degree. Next we explain how to apply the SO(3) decomposition to the statistical Navier-Stokes theory. We show how to extract information about the scaling behavior in the isotropic sector. Doing so furnishes a systematic way to assess the universality of the scaling exponents in this sector, clarifying the anisotropic origin of the many measurements that claimed the opposite. A systematic analysis of Direct Numerical Simulations and of experiments provides a strong support to the proposition that also for the non-linear problem there exists foliation of the statistical theory into sectors of the symmetry group. The exponents appear universal in each sector, and again strictly increasing as a function of the anisotropic degreee.Comment: 150 pages, 26 figures, submitted to Phys. Re

Statistical Properties of Turbulence: An Overview

We present an introductory overview of several challenging problems in the statistical characterisation of turbulence. We provide examples from fluid turbulence in three and two dimensions, from the turbulent advection of passive scalars, turbulence in the one-dimensional Burgers equation, and fluid turbulence in the presence of polymer additives.Comment: 34 pages, 31 figure

Statistical Theory of Magnetohydrodynamic Turbulence: Recent Results

In this review article we will describe recent developments in statistical theory of magnetohydrodynamic (MHD) turbulence. Kraichnan and Iroshnikov first proposed a phenomenology of MHD turbulence where Alfven time-scale dominates the dynamics, and the energy spectrum E(k) is proportional to k^{-3/2}. In the last decade, many numerical simulations show that spectral index is closer to 5/3, which is Kolmogorov's index for fluid turbulence. We review recent theoretical results based on anisotropy and Renormalization Groups which support Kolmogorov's scaling for MHD turbulence. Energy transfer among Fourier modes, energy flux, and shell-to-shell energy transfers are important quantities in MHD turbulence. We report recent numerical and field-theoretic results in this area. Role of these quantities in magnetic field amplification (dynamo) are also discussed. There are new insights into the role of magnetic helicity in turbulence evolution. Recent interesting results in intermittency, large-eddy simulations, and shell models of magnetohydrodynamics are also covered.Comment: 116 pages, Submitted to Phys. Rep., Due to size restrictions, the figures are not of high quality. Download from http://home.iitk.ac.in/~mk