Global expressions for high-order structure functions in Burgers turbulence

TL;DR

This paper derives exact, comprehensive expressions for high-order structure functions in Burgers turbulence, valid beyond the inertial range, facilitating better understanding and analysis of compressible turbulent systems.

Contribution

It introduces a novel derivation of exact structure function expressions that do not rely on shock statistics and are applicable beyond the inertial range.

Findings

Expressions are valid beyond the inertial range.

They link high-order structure functions without flow structure assumptions.

Results are easily verified through numerical simulations.

Abstract

Since the famous work by Kolmogorov on incompressible turbulence, the structure-function theory has been a key foundation of modern turbulence study. Due to the simplicity of Burgers turbulence, structure functions are calculated to arbitrary orders, which provides numerous implications for other compressible turbulent systems. We present the derivation of exact forcing-scale resolving expressions for high-order structure functions of the burgers turbulence. Compared with the previous theories where the structure functions are calculated in the inertial range based on the statistics of shocks, our expressions link high-order structure functions in different orders without extra information on the flow structure and are valid beyond the inertial range, therefore they are easily checked by numerical simulations.