Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations
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} $.
- Research Organization:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF), Oak Ridge, TN (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC)
- OSTI ID:
- 1565145
- Journal Information:
- Journal of Fluid Mechanics, Journal Name: Journal of Fluid Mechanics Vol. 732; ISSN 0022-1120
- Publisher:
- Cambridge University Press
- Country of Publication:
- United States
- Language:
- English
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