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

Abstract

We discuss continuous cascade models and their potential for modelling the energy dissipation in a turbulent flow. Continuous cascade processes, expressed in terms of stochastic integrals with respect to Lévy bases, are examples of ambit processes. These models are known to reproduce experimentally observed properties of turbulence: The scaling and self-scaling of the correlators of the energy dissipation and of the moments of the coarse-grained energy dissipation. We compare three models: a normal model, a normal inverse Gaussian model and a stable model. We show that the normal inverse Gaussian model is superior to both, the normal and the stable model, in terms of reproducing the distribution of the energy dissipation; and that the normal inverse Gaussian model is superior to the normal model and competitive with the stable model in terms of reproducing the self-scaling exponents. Furthermore, we show that the presented analysis is parsimonious in the sense that the self-scaling exponents are predicted from the one-point distribution of the energy dissipation, and that the shape of these distributions is independent of the Reynolds number. arXiv:1305.0690v2 [cond-mat.stat-mech]

Some Recent Developments in Ambit Stochastics

“…In [31] it has been shown that exponentials of trawl processes are able to reproduce the main stylized features of the (surrogate) energy dissipation observed for a wide range of datasets. Those stylized features include the one-dimensional marginal distributions and the scaling and self-scaling of the correlators.…”

“…In turbulence it is known (see the reference cited in [31]) that the correlator of the surrogate energy dissipation displays a scaling behaviour for a certain range of lags,…”

“…They are essentially universal in that they vary very little from one dataset to another, covering a large range of the so-called Reynolds numbers, a dimensionless quantity describing the character of the flow. In [31] the surrogate energy dissipation ε is, more specifically, modelled as…”

Incremental similarity and turbulence

“…[11]. And a recent study [10] of the Helium data discussed in Section 2, based on exponential LSS processes, has revealed a new type of universality for the energy dissipation.…”

“…For instance, Σ could be chosen to be an extIS class of processes σ 2 for which the log σ 2 are trawl process L (A + (t, 0)), as discussed in Section 3 In particular, L may be taken to be an N IG basis, as in [10].…”