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

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, $λ$, decrease with dimension, and $λ< 0$ 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.

Figures (5)

• Figure 1: Reynolds number dependence of $|\lambda T_{E0}|$, inset shows same data but in linear scale. For six dimensions, the Lyapunov exponent is always negative.
• Figure 2: QQ plot of normalized values. Low $Re$ for 3d is Gaussian (indicated by the straight line dependence), whereas 6d and high $Re$ 3d are not.
• Figure 3: Evolution of difference between two realisations with time, from early (light blue) to later (darker blue) times, showing different behavior with higher dimension: a) 3d $Re$ = 101, b) 4d $Re$ = 33.7, c) 5d $Re$ = 100, d) 6d $Re$ = 40.4, e) 3d $Re$ = 3.5.
• Figure 4: Steady state of difference at late times, showing reduced steady state of low wavenumber with higher dimension.
• Figure 5: Enstrophy spectrum, with the peak shifting to higher wavenumber with increasing spatial dimension.