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

Self-sustained oscillations in homogeneous shear flow

Generation of the large-scale coherent vortical structurs in homogeneous shear flow couples dynamical processes of energy and enstrophy production. In the large rate of strain limit, the simple estimates of the contributions to the energy and enstrophy equations result in a dynamical system, describing experimentally and numerically observed self-sustained non-linear oscillations of energy and enstrophy. It is shown that the period of these oscilaltions is independent upon the box size and the energy and enstrophy fluctuations are strongly correlated.Comment: 10 pages 6 figure

Mean- Field Approximation and Extended Self-Similarity in Turbulence

Recent experimental discovery of extended self-similarity (ESS) was one of the most interesting developments, enabling precise determination of the scaling exponents of fully developed turbulence. Here we show that the ESS is consistent with the Navier-Stokes equations, provided the pressure -gradient contributions are expressed in terms of velocity differences in the mean field approximation (Yakhot, Phys.Rev. E{\bf 63}, 026307, (2001)). A sufficient condition for extended self-similarity in a general dynamical systemComment: 8 pages, no figure

Kolmogorov turbulence in a random-force-driven Burgers equation: anomalous scaling and probability density functions

High-resolution numerical experiments, described in this work, show that velocity fluctuations governed by the one-dimensional Burgers equation driven by a white-in-time random noise with the spectrum $\overline{|f(k)|^2}\propto k^{-1}$ exhibit a biscaling behavior: All moments of velocity differences $S_{n\le 3}(r)=\overline{|u(x+r)-u(x)|^n}\equiv\overline{|\Delta u|^n}\propto r^{n/3}$, while $S_{n>3}\propto r^{\zeta_n}$ with $\zeta_n\approx 1$ for real $n>0$ (Chekhlov and Yakhot, Phys. Rev. E {\bf 51}, R2739, 1995). The probability density function, which is dominated by coherent shocks in the interval $\Delta u<0$, is ${\cal P}(\Delta u,r)\propto (\Delta u)^{-q}$ with $q\approx 4$.Comment: 12 pages, psfig macro, 4 figs in Postscript, accepted to Phys. Rev. E as a Brief Communicatio

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

Lattice Fluid Dynamics from Perfect Discretizations of Continuum Flows

We use renormalization group methods to derive equations of motion for large scale variables in fluid dynamics. The large scale variables are averages of the underlying continuum variables over cubic volumes, and naturally live on a lattice. The resulting lattice dynamics represents a perfect discretization of continuum physics, i.e. grid artifacts are completely eliminated. Perfect equations of motion are derived for static, slow flows of incompressible, viscous fluids. For Hagen-Poiseuille flow in a channel with square cross section the equations reduce to a perfect discretization of the Poisson equation for the velocity field with Dirichlet boundary conditions. The perfect large scale Poisson equation is used in a numerical simulation, and is shown to represent the continuum flow exactly. For non-square cross sections we use a numerical iterative procedure to derive flow equations that are approximately perfect.Comment: 25 pages, tex., using epsfig, minor changes, refernces adde

Turbulence without pressure in d dimensions

The randomly driven Navier-Stokes equation without pressure in d-dimensional space is considered as a model of strong turbulence in a compressible fluid. We derive a closed equation for the velocity-gradient probability density function. We find the asymptotics of this function for the case of the gradient velocity field (Burgers turbulence), and provide a numerical solution for the two-dimensional case. Application of these results to the velocity-difference probability density function is discussed.Comment: latex, 5 pages, revised and enlarge

A note on Burgers' turbulence

In this note the Polyakov equation [Phys. Rev. E {\bf 52} (1995) 6183] for the velocity-difference PDF, with the exciting force correlation function $\kappa (y)\sim1-y^{\alpha}$ is analyzed. Several solvable cases are considered, which are in a good agreement with available numerical results. Then it is shown how the method developed by A. Polyakov can be applied to turbulence with short-scale-correlated forces, a situation considered in models of self-organized criticality.Comment: 11 pages, Late