Dynamics of scaled norms of vorticity for the three-dimensional Navier-Stokes and Euler equations

Abstract

A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of $L^{2m}$-norms of vorticity ${\Omega }_m$ for $m\ge 1$. These are scaled to form the dimensionless sequence $D_m=({\varpi }_0^{-1}{\Omega }_m{)}^{{\alpha }_m}$ where ${\varpi }_0$ is a constant frequency and ${\alpha }_m=2m/(4m-3)$. A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of $D_m$ and $D_{m+1}$ while another is furnished by imposing a critical lower bound on ${\int }_0^t{D}_{m}d\tau$. The behaviour of the $D_m$ is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested.

arXiv commentTo appear in the Procedia IUTAM volume of papers Topological Fluid Dynamics