2015/06/09 by John Gibbon, J. D. Gibbon, Gibbon, J. D.
Engineering · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Computational Fluid Dynamics and Aerodynamics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis #nlin.CD
paper · pdf · doi:10.48550/arxiv.1506.03060
10 pages; 2 figures
openalex publication_date 2015/06/09 · arxiv created 2015/11/05 · arxiv updated 2015/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the 3D Navier-Stokes equations. A set of dimensionless frequencies \Ωm(t)\ are used which are based on L2m-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the Ωm (m>1) are slaved to Ω1 (the global enstrophy) in the form Ωm = Ω1Fm(Ω1). This is shaped by the constraint of two Hölder inequalities and a time average from which emerges a form for Fm which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions 1 ≤ λm(τ) ≤ 4, where curves of constant λm form the boundaries between tongue-shaped regions. In regions where 2.5 ≤ λm ≤ 4 and 1 ≤ λm ≤ 2 the Navier-Stokes equations are shown to be regular : numerical simulations appear to lie in the latter region. Only in the central region 2 < λm < 2.5 has no proof of regularity been found.