Scaling of the Maximum-Entropy Turbulence Energy Spectra

TL;DR

This paper demonstrates that a maximum-entropy derived turbulence energy spectrum can be accurately parameterized by turbulence variables, explaining inertial scaling and matching experimental data across different flow conditions.

Contribution

It introduces a parameterizable turbulence energy spectral function based on maximum entropy principles, linking spectral features to turbulence parameters and flow dimensionality.

Findings

Spectral function closely matches experimental data across scales.

Spectral width increases with Reynolds number, explaining inertial range scaling.

Applicable to various flow locations with local equilibrium achieved.

Abstract

The log-normal type of turbulence energy spectral function, derived from the maximum entropy principle, is shown to be parameterizable in terms of root turbulence variables including the Reynolds number. The spectral function is first compared with a number of experimental data sets, showing a very close agreement across the entire energy and length (wavenumber) scales. The peak wavenumber (m) and the width parameter (C2) prescribe the spectral location and broadening, respectively, when the Reynolds number increases, where C2 has ~ 1/Rem dependence. The energy scale is adjusted with a multiplicative factor. In this perspective, the inertial scaling from k-3 to k-5/3 when the flow transitions from two- to three-dimensions is explained as the increase in spectral width since the range of scales increases as Re1/6 and Re3/4 for two and three-dimensional turbulence, respectively. Energy spectra at various locations in channel flows are also reproduced using the same function, indicating applicability wherever local equilibrium is achieved. Therefore, based on a small number of scaling parameters the full energy spectra can be prescribed using the maximum-entropy formalism.