Extended Self Similarity works for the Burgers equation and why

TL;DR

This paper demonstrates that Extended Self-Similarity (ESS) improves scaling analysis in Burgers turbulence by reducing subdominant effects, providing a theoretical explanation for observed numerical enhancements, and suggesting potential applicability to 3D turbulence.

Contribution

The paper offers the first theoretical explanation for ESS's improved scaling in Burgers turbulence and proposes its potential relevance to three-dimensional turbulence.

Findings

ESS extends the scaling range by about three quarters of a decade.

Subdominant contributions to scaling are reduced in ESS compared to standard methods.

Theoretical explanation links ESS improvements to suppression of subdominant effects.

Abstract

Extended Self-Similarity (ESS), a procedure that remarkably extends the range of scaling for structure functions in Navier--Stokes turbulence and thus allows improved determination of intermittency exponents, has never been fully explained. We show that ESS applies to Burgers turbulence at high Reynolds numbers and we give the theoretical explanation of the numerically observed improved scaling at both the infrared and ultraviolet end, in total a gain of about three quarters of a decade: there is a reduction of subdominant contributions to scaling when going from the standard structure function representation to the ESS representation. We conjecture that a similar situation holds for three-dimensional incompressible turbulence and suggest ways of capturing subdominant contributions to scaling.