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

Michele V Bartuccelli, Guido Gentile

TL;DR

This paper develops an analytical criterion for detecting intermittent turbulent behavior in PDE solutions by employing the crest factor $C_f = \|u\|_{\infty} / J_0^{1/2}$. It shows that linear PDEs (heat, wave, Stokes) yield time-invariant or decaying crest factors, implying no intermittent turbulence, and that nonlinear cases like Burgers’ equation likewise lack turbulence under the tested setups. For incompressible Navier–Stokes equations, the authors derive bounds on time-averaged crest factors, linking large intermittency to significant fluctuations in $\|Du\|_{\infty}$ and connecting to Kolmogorov-type dissipation scales in 3D, while 2D on the torus exhibits growth tied to forcing scales and viscosity but not strong intermittency in typical regimes. Overall, the crest-factor criterion provides a quantitative, time-resolved diagnostic to distinguish mild, hard, and intermittent turbulence and offers a practical tool for analyzing PDE simulations. The work emphasizes how CF analysis can guide interpretation of numerical results and deepen understanding of turbulence in fluid dynamics models.

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.

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

TL;DR

This paper develops an analytical criterion for detecting intermittent turbulent behavior in PDE solutions by employing the crest factor . It shows that linear PDEs (heat, wave, Stokes) yield time-invariant or decaying crest factors, implying no intermittent turbulence, and that nonlinear cases like Burgers’ equation likewise lack turbulence under the tested setups. For incompressible Navier–Stokes equations, the authors derive bounds on time-averaged crest factors, linking large intermittency to significant fluctuations in and connecting to Kolmogorov-type dissipation scales in 3D, while 2D on the torus exhibits growth tied to forcing scales and viscosity but not strong intermittency in typical regimes. Overall, the crest-factor criterion provides a quantitative, time-resolved diagnostic to distinguish mild, hard, and intermittent turbulence and offers a practical tool for analyzing PDE simulations. The work emphasizes how CF analysis can guide interpretation of numerical results and deepen understanding of turbulence in fluid dynamics models.

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.
Paper Structure (15 sections, 3 theorems, 86 equations)

This paper contains 15 sections, 3 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. The Crest Factor
  3. Basic Definitions and Setup
  4. The Crest Factor and Analysis of the Solutions
  5. Linear PDEs
  6. The Heat Equation
  7. The Wave Equation
  8. The time-dependent Stokes problem
  9. Nonlinear PDEs
  10. Burgers' Equation
  11. Incompressible Navier-Stokes Equations
  12. Stokes Second Problem
  13. Motion in a Half-Plane
  14. Incompressible Navier-Stokes Equations on the Torus
  15. Conclusions

Key Result

Theorem 4.1

For small values of $\nu$, the long time-averaged crest factor for the two-dimensional incompressible NSE on the torus $[0,2\pi]^2$ obeys the estimate where with $L=2\pi$, $\eta \approx 1.82$, and $c_1$ and $c_2$ two positive constants such that $c_1 + c_2 = 1$.

Theorems & Definitions (3)

  • Theorem 4.1
  • Lemma
  • Theorem 4.2