On an Analytical Criterion for Detecting Intermittent Turbulent Behaviour of Solutions of Partial Differential Equations

TL;DR

This paper introduces an analytical criterion based on the crest factor to identify intermittent turbulent behavior in solutions of partial differential equations, aiding understanding of turbulence phenomena.

Contribution

It proposes a novel crest factor-based criterion to distinguish between turbulent and non-turbulent solutions of PDEs, validated on classical equations.

Findings

The crest factor effectively detects intermittent turbulence.

The criterion differentiates between stationary and non-stationary turbulence.

Application to classical PDEs demonstrates practical utility.

Abstract

A main question in the study of partial differential equations is the following: how do we understand the nature of the solutions and, in particular, how do we determine if a given solution shows turbulent or non-turbulent behaviour? Being able to answer such a question would be a major advance in the comprehension of the nature of turbulence. In this paper we focus on the case of intermittent turbulence and provide an analytical criterion, based on the crest factor, which captures the essential feature of the solutions. By computing the crest factor for the solutions of some classical equations, both linear and nonlinear, we illustrate the capability of the criterion for discerning between solutions exhibiting time-intermittent turbulence behaviour and solutions which either are not turbulent or show statistically stationary turbulence, like, for example, in the case described by Kolmogorov's theory.