Vorticity moments in four numerical simulations of the 3D Navier-Stokes equations

Abstract

The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations on a periodic box $[0,\,L]^{3}$ is addressed through four sets of numerical simulations that calculate a new set of variables defined by $D_{m}(t) = \left(\varpi_{0}^{-1}\Omega_{m}\right)^{\alpha_{m}}$ for $1 \leq m \leq \infty$ where $\alpha_{m}= \frac{2m}{4m-3}$ and $\left[\Omega_{m}(t)\right]^{2m} = L^{-3}\I |\bom|^{2m}dV$ with $\varpi_{0} = \nu L^{-2}$. All four simulations unexpectedly show that the $D_{m}$ are ordered for $m = 1\,,...,\,9$ such that $D_{m+1} < D_{m}$. Moreover, the $D_{m}$ squeeze together such that $D_{m+1}/D_{m}\nearrow 1$ as $m$ increases. The first simulation is of very anisotropic decaying turbulence\,; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at constant Grashof number respectively\,; the fourth is of very high Reynolds number forced, stationary, isotropic turbulence at up to resolutions of $4096^{3}$.