Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions

TL;DR

This paper compares different theoretical approaches to describing multi-scale velocity correlations in turbulence, demonstrating their connections, and extends multifractal predictions to velocity gradient correlations, validated by numerical simulations.

Contribution

It establishes the link between fusion rules and Markov processes in turbulence and generalizes multifractal predictions to higher-order velocity gradient correlations.

Findings

Fusion rules follow from Markov property under inertial range scaling.

Derived a generalized multifractal prediction for velocity gradient correlations.

Validated theoretical results with direct numerical simulations of Burgers turbulence.

Abstract

We compare different approaches towards an effective description of multi-scale velocity field correlations in turbulence. Predictions made by the operator product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal (MF) approach derived in [Phys. Rev. Lett. 80, 3244 (1998)] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from "first principles" in the case of Burgers equation. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence.