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

2012/12/04 by J. D. Gibbon, Gibbon, J. D.
Mathematics · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #nlin.CD

paper · pdf · doi:10.48550/arxiv.1212.0684

To appear in the Procedia IUTAM volume of papers Topological Fluid Dynamics

arxiv created 2012/12/04 · arxiv updated 2012/12/05

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 Ω_m for m≥ 1. These are scaled to form the dimensionless sequence D_m= (\varpi₀^-1Ω_m)^α_m where \varpi₀ is a constant frequency and α_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 ∫₀^tD_m dτ. 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.

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