Generalized Knudsen number for unsteady fluid flow

We explore the scaling behavior of an unsteady flow that is generated by an oscillating body of finite size in a gas. If the gas is gradually rarefied, the Navier-Stokes equations begin to fail and a kinetic description of the flow becomes more appropriate. The failure of the Navier-Stokes equations can be thought to take place via two different physical mechanisms: either the continuum hypothesis breaks down as a result of a finite size effect or local equilibrium is violated due to the high rate of strain. By independently tuning the relevant linear dimension and the frequency of the oscillating body, we can experimentally observe these two different physical mechanisms. All the experimental data, however, can be collapsed using a single dimensionless scaling parameter that combines the relevant linear dimension and the frequency of the body. This proposed Knudsen number for an unsteady flow is rooted in a fundamental symmetry principle, namely, Galilean invariance

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

Anomalous Scaling of Structure Functions and Dynamic Constraints on Turbulence Simulations

The connection between anomalous scaling of structure functions (intermittency) and numerical methods for turbulence simulations is discussed. It is argued that the computational work for direct numerical simulations (DNS) of fully developed turbulence increases as $Re^{4}$, and not as $Re^{3}$ expected from Kolmogorov's theory, where $Re$ is a large-scale Reynolds number. Various relations for the moments of acceleration and velocity derivatives are derived. An infinite set of exact constraints on dynamically consistent subgrid models for Large Eddy Simulations (LES) is derived from the Navier-Stokes equations, and some problems of principle associated with existing LES models are highlighted.Comment: 18 page

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

The dynamics of velocity fluctuations, governed by the one-dimensional Burgers equation, driven by a white-in-time random force with the spatial spectrum \overline{|f(k)|^2}\proptok^{-1}, is considered. High-resolution numerical experiments conducted in this work give the energy spectrum $E(k)\propto k^{-\beta}$ with $\beta =5/3\pm 0.02$. The observed two-point correlation function $C(k,\omega)$ reveals $\omega\propto k^z$ with the "dynamical exponent" $z\approx 2/3$. High-order moments of velocity differences show strong intermittency and are dominated by powerful large-scale shocks. The results are compared with predictions of the one-loop renormalized perturbation expansion.Comment: 13 LaTeX pages, psfig.sty macros, Phys. Rev. E 51, R2739 (1995)

Closure of two dimensional turbulence: the role of pressure gradients

Inverse energy cascade regime of two dimensional turbulence is investigated by means of high resolution numerical simulations. Numerical computations of conditional averages of transverse pressure gradient increments are found to be compatible with a recently proposed self-consistent Gaussian model. An analogous low order closure model for the longitudinal pressure gradient is proposed and its validity is numerically examined. In this case numerical evidence for the presence of higher order terms in the closure is found. The fundamental role of conditional statistics between longitudinal and transverse components is highlighted.Comment: 4 pages, 2 figures, in press on PR

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