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Bounds for Euler from vorticity moments and line divergence

  • Robert M. Kerr (a1)

The inviscid growth of a range of vorticity moments is compared using Euler calculations of anti-parallel vortices with a new initial condition. The primary goal is to understand the role of nonlinearity in the generation of a new hierarchy of rescaled vorticity moments in Navier–Stokes calculations where the rescaled moments obey ${D}_{m} \geq {D}_{m+ 1} $ , the reverse of the usual ${\Omega }_{m+ 1} \geq {\Omega }_{m} $ Hölder ordering of the original moments. Two temporal phases have been identified for the Euler calculations. In the first phase the $1\lt m\lt \infty $ vorticity moments are ordered in a manner consistent with the new Navier–Stokes hierarchy and grow in a manner that skirts the lower edge of possible singular growth with ${ D}_{m}^{2} \rightarrow \sup \vert \boldsymbol{\omega} \vert \sim A_{m}{({T}_{c} - t)}^{- 1} $ where the ${A}_{m} $ are nearly independent of $m$ . In the second phase, the new ${D}_{m} $ ordering breaks down as the ${\Omega }_{m} $ converge towards the same super-exponential growth for all $m$ . The transition is identified using new inequalities for the upper bounds for the $- \mathrm{d} { D}_{m}^{- 2} / \mathrm{d} t$ that are based solely upon the ratios ${D}_{m+ 1} / {D}_{m} $ , and the convergent super-exponential growth is shown by plotting $\log (\mathrm{d} \log {\Omega }_{m} / \mathrm{d} t)$ . Three-dimensional graphics show significant divergence of the vortex lines during the second phase, which could be what inhibits the initial power-law growth.

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Beale, J. T, Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions of the 3-D Euler equations. Commun. Math. Phys. 94, 6166.
Bustamante, M. D. & Kerr, R. M. 2008 3D Euler about a 2D symmetry plane. Physica D 237, 19121920.
Deng, J., Hou, T. Y. & Yu, X. 2005 Geometric properties and nonblowup of 3D incompressible Euler flow. Comm. Part. Diff. Equ. 30, 225243.
Doering, C. R. 2009 The 3D Navier–Stokes problem. Annu. Rev. Fluid. Mech. 41, 109128.
Donzis, D., Gibbon, J. D., Kerr, R. M., Pandit, R., Gupta, A. & Vincenzi, D. 2013 Rescaled vorticity moments in the 3D Navier–Stokes equations. J. Fluid Mech. (submitted), arXiv:1302.1768.
Gibbon, J. D. 2012 Conditional regularity of solutions of the three-dimensional Navier–Stokes equations and implications for intermittency. J. Math. Phys. 53, 115608.
Gibbon, J. D. 2013 Dynamics of scaled norms of vorticity for the three-dimensional Navier–Stokes and Euler equations. Procedia IUTAM 7, 4958.
Hou, T. Y. 2008 Blow-up or no blow-up? The interplay between theory and numerics. Physica D 237, 19371944.
Kerr, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A 5, 17251746.
Kerr, R. M. 1996 Cover illustration: vortex structure of Euler collapse. Nonlinearity 9, 271272.
Kerr, R. M. 2011 Vortex stretching as a mechanism for quantum kinetic energy decay. Phys. Rev. Lett. 106, 224501.
Kerr, R. M. 2012 Dissipation and enstrophy statistics in turbulence: are the simulations and mathematics converging? J. Fluid Mech. 700, 14.
Kerr, R. M. 2013 Swirling, turbulent vortex rings formed from a chain reaction of reconnection events. Phys. Fluids 25, 065101.
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2012 Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers. J. Fluid Mech. 700, 515.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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