Effect of spatial dimension on a model of fluid turbulence

TL;DR

This study explores how increasing spatial dimensions affect fluid turbulence characteristics using numerical simulations, revealing enhanced energy transfer and dissipation with higher dimensions, and suggesting new insights into turbulence behavior in high-dimensional spaces.

Contribution

The paper derives relationships for structure functions and spectra in arbitrary dimensions and compares these with four-dimensional simulations, providing new understanding of turbulence in higher dimensions.

Findings

Energy spectra exhibit a growing bottleneck effect with dimension.

Transfer spectra show a peak shift indicating increased forward energy transfer.

Dissipation rate increases with dimension, and enstrophy production peaks around five dimensions.

Abstract

A numerical study of the $d$-dimensional Eddy Damped Quasi-Normal Markovian equations is performed to investigate the dependence on spatial dimension of homogeneous isotropic fluid turbulence. Relationships between structure functions and energy and transfer spectra are derived for the $d$-dimensional case. Additionally, an equation for the $d$-dimensional enstrophy analogue is derived and related to the velocity derivative skewness. Comparisons are made to recent four dimensional direct numerical simulation results. Measured energy spectra show a magnified bottleneck effect which grows with dimension whilst transfer spectra show a varying peak in the non-linear energy transfer as the dimension is increased. These results are consistent with an increased forward energy transfer at higher dimensions, further evidenced by measurements of a larger asymptotic dissipation rate with growing dimension. The enstrophy production term, related to the velocity derivative skewness, is seen to reach a maximum at around five dimensions and may reach zero in the limit of infinite dimensions, raising interesting questions about the nature of turbulence in this limit.