Measuring the effect of spatial dimension on hydrodynamic turbulence using direct numerical simulation

TL;DR

This study uses direct numerical simulations to explore how increasing spatial dimensions affect turbulence characteristics, revealing decreased chaos, non-Gaussian statistics, and altered vortex dynamics in higher dimensions.

Contribution

It provides new insights into the impact of spatial dimension on turbulence behavior, especially in six dimensions where chaos diminishes but turbulence-like features persist.

Findings

Lyapunov exponents decrease with dimension

Turbulence exhibits non-Gaussian statistics in six dimensions

Small-scale perturbations do not influence large scales

Abstract

We perform direct numerical simulation of the incompressible Navier-Stokes equation with forcing at different spatial dimensions and measure turbulent and chaotic properties. Lyapunov exponents, $\lambda$, decrease with dimension, and $\lambda < 0$ for all simulations in six-dimensions up to $Re = 40$. These six-dimensional simulations display non-Gaussian statistics and other behavior similar to well developed turbulence despite their lack of chaos. Further, we find that small scale perturbations do not extend to the largest scales and that this terminal scale between correlation and decorrelation shrinks with dimension. We theorize that this change is related to the increased rate of vortex stretching. We find the interplay between turbulent and chaotic properties changes with increasing dimension.