Turbulence and Stochastic Processes: Kolmogorov's Ideas 50 Years On . Edited by J. C. R. H UNT , O. M. P HILLIPS and D. W ILLIAMS . Proceedings of the Royal Society, London , A, vol. 434, 1991, pp. 1–240. £19.50 (original) (raw)
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The Kolmogorov–Obukhov Statistical Theory of Turbulence
Journal of Nonlinear Science, 2013
In 1941 Kolmogorov and Obukhov proposed that there exists a statistical theory of turbulence that should allow the computation of all the statistical quantities that can be computed and measured in turbulent systems. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier-Stokes equation. The additive noise in the stochastic Navier-Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov '62 scaling hypothesis with the She-Leveque intermittency corrections. Then we compute the invariant measure of turbulence writing the stochastic Navier-Stokes equation as an infinite-dimensional Ito process and solving the linear Kolmogorov-Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.
“Locally homogeneous turbulence”: Is it an inconsistent framework?
Physics of Fluids, 2005
In his first 1941 paper Kolmogorov assumed that the velocity has increments which are homogeneous and independent of the velocity at a suitable reference point. This assumption of local homogeneity is consistent with the nonlinear dynamics only in an asymptotic sense when the reference point is far away. The inconsistency is illustrated numerically using the Burgers equation. Kolmogorov's derivation of the four-fifths law for the third-order structure function and its anisotropic generalization are actually valid only for homogeneous turbulence, but a local version due to Duchon and Robert still holds. A Kolomogorov-Landau approach is proposed to handle the effect of fluctuations in the large-scale velocity on small-scale statistical properties; it is is only a mild extension of the 1941 theory and does not incorporate intermittency effects.
Theoretical and Mathematical Physics, 1992
A new approach to the description of fully developed turbulence is developed on the basis of the maximum entropy principle and the renormalization group. The Kolmogorov dimension for the velocity field is obtained, and a conjecture which explains the experimentally observed deviations from this dimension is proposed. The calculated anomalous dimension of the energy dissipation operator differs from the prediction of Kolmogorov's theory.
Convection and Chaos in Fluids, 1987
Experimental data from a turbulent jet flow is analysed in terms of an additive, continuous stochastic process where the usual time variable is replaced by the scale. We show that the energy transfer through scales is well described by a linear Langevin equation, and discuss the statistical properties of the corresponding random force in detail. We find that the autocorrelation function of the random force decays rapidly: the process is therefore Markov for scales larger than Kolmogorov's dissipation scale η. The corresponding autocorrelation scale is identified as the elementary step of the energy cascade. However, the probability distribution function of the random force is both non-Gaussian and weakly scale-dependent.
On the behavior of homogeneous, isotropic and stationary turbulence
The recent development of a statistical model for incompressible Navier-Stokes (NS) fluids based on inverse kinetic theory (IKT, 2004(IKT, -2008 poses the problem of searching for particular realizations of the theory which may be relevant for the statistical description of turbulence and in particular for the so-called homogeneous, isotropic and stationary turbulence (HIST). Here the problem is set in terms of the 1−point velocity probability density function (PDF) which determines a complete IKT-statistical model for NS fluids. This raises the interesting question of identifying the statistical assumptions under which a Gaussian PDF can be achieved in such a context. In this paper it is proven that for the IKT statistical model, HIST requires necessarily that f 1 must be SIED (namely stationary, isotropic and everywhere-defined). This implies, in turn, that the functional form of the PDF is uniquely prescribed at all times. In particular, it is found that necessarily the PDF must coincide with an isotropic Gaussian distribution. The conclusion is relevant for the investigation of the so-called homogenous, isotropic and stationary turbulence.
Experimental indications for Markov properties of small-scale turbulence
Journal of Fluid Mechanics, 2001
We present a stochastic analysis of a data set consisting of 1.25 × 107 samples of the local velocity measured in the turbulent region of a round free jet. We find evidence that the statistics of the longitudinal velocity increment v(r) can be described as a Markov process. This new approach to characterize small-scale turbulence leads to a Fokker–Planck equation for the r-evolution of the probability density function (p.d.f.) of v(r). This equation for p(v, r) is completely determined by two coefficients D1(v, r) and D2(v, r) (drift and diffusion coefficient, respectively). It is shown how these coefficients can be estimated directly from the experimental data without using any assumptions or models for the underlying stochastic process. The solutions of the resulting Fokker–Planck equation are compared with experimentally determined probability density functions. It is shown that the Fokker–Planck equation describes the measured p.d.f.(s) correctly, including intermittency effects...
Small-scale anisotropy in Lagrangian turbulence
New Journal of Physics, 2006
We report measurements of the second-order Lagrangian structure function and the Lagrangian velocity spectrum in an intensely turbulent laboratory flow. We find that the asymmetries of the large-scale flow are reflected in the small-scale statistics. In addition, we present new measurements of the Lagrangian structure function scaling constant C 0 , which is of central importance to stochastic turbulence models as well as to the understanding of turbulent pair dispersion and scalar mixing. The scaling of C 0 with the turbulence level is also investigated, and found to be in agreement with an existing model.