On A Discrete Dynamical Model For Local Turbulence

Journal of Fluid Mechanics, 1978

We present a phenomenological model of intermittency called the P-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales N &2-" only a fraction /3n of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974)' is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turnover times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation. The P-model leads in an elementary way to the concept of the self-similarity dimension D , a special case of Mandelbrot's (1974, 1976) 'fractal dimension'. For threedimensional turbulence, the correction B to the Q exponent of the energy spectrum is equal to +(3-D) and is related to the exponent p of the dissipation correlation function by B = Qp (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality B < Qp. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second-and fourth-order moments have the same viscous cutoff. The predictions of the P-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function (8v6(Z)) and the dissipation correlation function (e(r) e(r + 1)) are related by (Svs(l))/Z2 N +(r) e(r + 1)) in both models. We conjecture that this relation is model independent. intermittency are indicated. Finally, some possible directions for further numerics? and experimental work on

Emergent extreme events are a key characteristic of complex dynamical systems. The main tool for detailed and deep understanding of their stochastic dynamics is the statistics of time intervals of extreme events. Analyzing extensive experimental data, we demonstrate that for the velocity time series of fully-developed turbulent flows, generated by (i) a regular grid; (ii) a cylinder; (iii) a free jet of helium, and (iv) a free jet of air with the Taylor Reynolds numbers Re λ from 166 to 893, the interoccurrence time distributions P(τ) above a positive threshold Q in the inertial range is described by a universal q-exponential function, P(τ) = β(2 − q)[1 − β(1 − q)τ] 1/(1−q) , which may be due to the superstatistical nature of the occurrence of extreme events. Our analysis provides a universal description of extreme events in turbulent flows. Dynamical systems in biology, physics, chemistry, computer science, social and other types of networks, and many other fields and phenomena exhibit rich and complex behavior that are not amenable to the classical methods of analysis. Although there is no consensus on what constitutes complex systems, they are usually referred to as such because they typically consist of many subparts that interact with each other over many length and/or time scales, not uniquely but in several ways. Due to their complexity, the difficulty in precisely describing their behavior and properties, and the ubiquity of complex dynamical systems in nature, one approach to the analysis of such systems has been based on trying to identify their universal features. That is, one tries to discover whether there are certain features of complex dynamical systems that are exhibited by a large class of them, regardless of the details of their structure. If such universal features can be identified, then one can utilize them to develop the simplest model of such systems that exhibit the universal properties and utilize it to gain further insights into their dynamics. One of the main features of experimental data for complex dynamical systems is their small time scale inter-mittency. Intuitively, intermittency represents the strongly correlated fluctuations that lead to deviations of increments probability distribution functions (PDF) from Gaussian statistics at small scales and is defined by, δx r = x(t + r) − x(t), for a given small r. The stretched exponential like behavior of P(δx r) changes as the scale r increases from the smallest to the largest ones. A prototype of complex dynamical systems is turbulent flow. As is well understood, any flow field is the result of the competition between two factors, namely, the inertial and viscous forces 1. The Taylor Reynolds numbers is defined as Re λ = U rms λ/ν, where U rms , λ, and ν are, respectively, the root mean-square of the fluctuations in the velocity field, the Taylor length scale of the flow field, and the kinematic viscosity 2. In turbulent flows, which correspond to large Re λ , i.e. the dominance of inertial over viscous forces, the fluid particles follow complex trajecto-ries resembling stochastic motion. The flow is characterized by an energy flux cascade from a large integral scale L towards the intermediate and eventually smallest scales in which energy is dissipated by molecular viscosity. The inertial range is defined by the scales λ < l < L. Traditionally, the universality in fully-developed turbulence is expressed in terms of the multiscaling exponents that describe the multiscaling of the positive ordered structure functions of the velocity increments in the inertial range under the steady state condition. More precisely, the longitudinal velocity increments, δ = + − ⋅ v v x r v x (() ()) l r r , across a distance r (in the inertial range) is believed to behave as, δ ζ

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...

Physical Review E, 2010

A novel model of intermittency is presented in which the dynamics of the rates of energy transfer between successive steps in the energy cascade is described by a hierarchy of stochastic differential equations. The probability distribution of velocity increments is calculated explicitly and expressed in terms of generalized hypergeometric functions of the type n F 0 , which exhibit power-law tails. The model predictions are found to be in good agreement with experiments on a low temperature gaseous helium jet. It is argued that distributions based on the functions n F 0 might be relevant also for other physical systems with multiscale dynamics.

Scientia Marina

This paper provides a short introduction to the consequences of intermittency on the statistical properties of fully developed turbulence, mainly on (a) scaling laws for the different moments of velocity, (b) energy distribution and (c) diffusion behaviour. The description of intermittency is carried out in the fractal β model and in a more general multifractal perspective.