Stochastic energy-cascade model for (1+1)-dimensional fully developed turbulence

Abstract

Geometrical random multiplicative cascade processes are often used to model positivevalued multifractal fields such as the energy dissipation in fully developed turbulence. We propose a dynamical generalization describing the energy dissipation in terms of a continuous and homogeneous stochastic field in one space and one time dimension. In the model, correlations originate in the overlap of the respective spacetime histories of field amplitudes. The theoretical two-and three-point correlation functions are found to be in good agreement with their equal-time counterparts extracted from wind tunnel turbulent shear flow data.

Ambit Processes; with Applications to Turbulence and Tumour Growth

“…Relation (52) has been applied to turbulent data in Schmiegel et al (2004) where the shape of the ambit set has been extracted from scaling two-point correlators. As a consequence the higher order correlators are fixed and the three-point correlators have been successfully compared to experimental data.…”

“…The concept of ambit processes discussed in this paper arose out of a current study (Barndorff-Nielsen and Schmiegel (2005), Schmiegel et al (2006), Barndorff-Nielsen and Schmiegel (2004), Schmiegel et al (2004) and Schmiegel (2005a)) the ultimate aim of which is to build a realistic stochastic process model of 3-dimensional turbulent velocity fields, in the spirit of Kolmogorov's phenomenological theory (Frisch (1995)) -and beyond. Besides applications to turbulence, the concept has also been used in modelling the growth of cancer tumours (Schmiegel (2005b)), and it should be of interest to other fields as well.…”

A dissipative random velocity field for fully developed fluid turbulence

“…Let us also mention the iterative procedure of Rosales & Meneveau (2008) that gives a realistic picture but which is not explicit, making analytical results out of reach at the present time. In a one-dimensional context, several models have been proposed in the literature in order to apply the discrete cascade models to reproduce synthetically the observed fluctuations of longitudinal velocity profiles, including a model for energy transfer (Juneja et al 1994) with additional parameters and propositions to extend to spatio-temporal (Biferale et al 1998) and Levy-based (Schmiegel et al 2004) stochastic representations. In a different spirit, Nawroth & Peinke (2006) propose to reconstruct velocity time series, starting from a time series at a given (small) scale and assuming a Markov property in scale.…”

A causal continuous-time stochastic model for the turbulent energy cascade in a helium jet flow

“…Choosing the law of the cascade generators to be log-normal yields the Kolmogorov-Oboukhov model. A continuous analogue to the discrete multiplicative cascade processes is formulated in terms of integrals with respect to Lévy bases and has been shown [4,39,41] to be computationally tractable and to accurately describe the two-and three-point statistics of the energy dissipation.…”