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

JFM classification

Turbulent Flows: Intermittency Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 732 , 10 October 2013 , pp. 316 - 331
Copyright
©2013 Cambridge University Press 

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