Parametric modelling of turbulence

Abstract

Some steps are taken towards a parametric statistical model for the velocity and velocity derivative fields in stationary turbulence, building on the background of existing theoretical and empirical knowledge of such fields. While the ultimate goal is a model for the three-dimensional velocity components, and hence for the corresponding velocity derivatives, we concentrate here on the stream wise velocity component. Discrete and continuous time stochastic processes of the first-order autoregressive type and with one-dimensional marginals having log-linear tails are constructed and compared with two large data-sets. It turns out that a first-order autoregression that fits the local correlation structure well is not capable of describing the correlations over longer ranges. A good fit locally as well as at longer ranges is achieved by using a process that is the sum of two independent autoregressions. We study this type of model in some detail. We also consider a model derived from the above-mentioned autoregressions and with dependence structure on the borderline to long-range dependence. This model is obtained by means of a general method for construction of processes with long-range dependence. Some suggestions for future empirical and theoretical work are given.

The Kolmogorov–Obukhov Statistical Theory of Turbulence

“…The transition to turbulence occurs at R ∼ 500 and the 3 flow is typically fully turbulent when R ∼ 2000. Most flows occuring in nature are turbulent even a small stream can have Reynolds number of 10 4 and for a large river it is not unusual that R ∼ 10 6 . The deterministic Navier-Stokes equation describes laminar flow that may exist when the Reynolds number is large, but then laminar flow is usually unstable.…”

“…ε is the dissipation rate (6). The coefficient C is a constant times a Sobolov space norm of u, by the estimate (13), see [12].…”

“…Because of intermittency each structure function generates its own NIG distribution with separate parameters. Then we compare these PDFs with DNS results and experimental data, see also [6,7].…”

Estimating functions for diffusion-type processes

“…An autocorrelation function of the form Sørensen (1990) and Bibby, Skovgaard & Sørensen (2005). A simple model with autocorrelation function of the form (7.21) is the sum of diffusions…”

Methods and Techniques of Complex Systems Science: An Overview

“…It is often possible to describe a nonlinear dynamical system through an effective linear statistical model, provided the nonlinearities are cooperative enough to appear as noise [61]. It is an under-appreciated fact that this is at least sometimes true even of turbulent flows [62,63]; the generality of such an approach is not known. Certainly, if you care only about predicting a time series, and not about its structure, it is always a good idea to try a linear model first, even if you know that the real dynamics are highly nonlinear.…”