Symmetry and Hamiltonian structure of the scaling equation in isotropic turbulence

TL;DR

This paper revisits the scaling equation in isotropic turbulence, analyzing its symmetry and Hamiltonian structure to deepen understanding of turbulence features like spectra and large-scale dynamics.

Contribution

It provides a comprehensive analysis of the scaling equation's symmetry and Hamiltonian structure, offering new insights into turbulence behavior beyond previous studies.

Findings

New exact solutions for the scaling equation

Insights into turbulence spectra and large-scale dynamics

Enhanced understanding of self-preservation in isotropic turbulence

Abstract

The assumption of similarity and self-preservation, which permits an analytical determination of the energy decay in isotropic turbulence, has played an important role in the development of turbulence theory for more than half a century. Sedov (1944), who first found an ingenious way to obtain two equations from one. Nonethless, it appears that this problem has never been reinvestigated in depth subsequent to this earlier work. In the present paper, such an analysis is carried out, yielding a much more complete picture of self-preservation isotropic turbulence. Based on these exact solutions, some physically significant consequences of recent advances in the theory of self-preserved homogenous statistical solution of the Navier-Stokes equations are presented. New results could be obtained for the analysis on turbulence features, such as the scaling behavior, the spectrum, and also the large scale dynamics. The general energy spectra and their behavior in different wave number range are investigated. This letter only focus on the scaling equation.