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

Published online by Cambridge University Press:  04 September 2013

Diego A. Donzis
Department of Aerospace Engineering, Texas A&M University, College Station, Texas, TX 77840, USA
John D. Gibbon*
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Anupam Gupta
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
Robert M. Kerr
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK
Rahul Pandit
Department of Physics, Indian Institute of Science, Bangalore 560 012, India Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India
Dario Vincenzi
Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France
Email address for correspondence:


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)= {({ \varpi }_{0}^{- 1} {\Omega }_{m} )}^{{\alpha }_{m} } $ for $1\leq m\leq \infty $ where ${\alpha }_{m} = 2m/ (4m- 3)$ and ${[{\Omega }_{m} (t)] }^{2m} = {L}^{- 3} \int \nolimits _{\mathscr{V}} {\vert \boldsymbol{\omega} \vert }^{2m} \hspace{0.167em} \mathrm{d} V$ with ${\varpi }_{0} = \nu {L}^{- 2} $. All four simulations unexpectedly show that the ${D}_{m} $ are ordered for $m= 1, \ldots , 9$ such that ${D}_{m+ 1} \lt {D}_{m} $. Moreover, the ${D}_{m} $ squeeze together such that ${D}_{m+ 1} / {D}_{m} \nearrow 1$ as $m$ increases. The values of ${D}_{1} $ lie far above the values of the rest of the ${D}_{m} $, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. 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 fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of $409{6}^{3} $.

©2013 Cambridge University Press 

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