Vorticity moments in four numerical simulations of the 3D Navier-Stokes equations
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abstract
AbstractThe issue of intermittency in numerical solutions of the 3D NavierStokes 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 $1leq mleq infty $ where ${alpha }_{m} = 2m/ (4m- 3)$ and ${[{Omega }_{m} (t)] }^{2m} = {L}^{- 3} int olimits _{mathscr{V}} {vert \boldsymbol{omega} vert }^{2m} hspace{0.167em} mathrm{d} V$ with ${varpi }_{0} = u {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 ${D}_{m+ 1} / {D}_{m} earrow 1$ 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 NavierStokes 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} $.
