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 $1\le m\le \infty$ where ${\alpha }_m=\frac{2m}{4m-3}$ and $\left[Ω_{m}(t)\right]^{2m} = L^{-3}\I |\bom|^{2m}dV$ 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 $4096^3$.

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