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Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

Hou, Thomas Y. and Li, Ruo (2006) Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations. Journal of Nonlinear Science, 16 (6). pp. 639-664. ISSN 0938-8974. http://resolver.caltech.edu/CaltechAUTHORS:20160322-090700507

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Abstract

We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition [15] [Phys. Fluids 5 (1993), 1725]. We use a pseudo-spectral method with resolution up to 1536 × 1024 × 3072 to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than doubly exponential in time, up to t = 19, beyond the singularity time t = 18.7 predicted by Kerr's computations [15], [22]. The velocity, the enstrophy, and the enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00332-006-0800-3DOIArticle
https://rdcu.be/bOWpPPublisherFree ReadCube access
https://arxiv.org/abs/math-ph/0602051arXivDiscussion Paper
Additional Information:© 2006 Springer. Received February 20, 2006; accepted April 27, 2006; Online publication August 25, 2006. We would like to thank Prof. Lin-Bo Zhang from the Institute of Computational Mathematics in Chinese Academy of Sciences (CAS) for providing us with the computing resource to perform this large-scale computational project. Additional computing resource was provided by the Center of High Performance Computing in CAS. We also thank Prof. Robert Kerr for providing us with his Fortran subroutine that generated his initial data. This work was supported in part by NSF under the NSF FRG grant DMS-0353838 and ITR Grant ACI-0204932. Part of this work was done while Hou visited the Academy of Systems and Mathematical Sciences of CAS in the summer of 2005 as a member of the Oversea Outstanding Research Team for Complex Systems. Finally, we would like to thank Profs. Hector Ceniceros, Charles Fefferman, and Robert Kerr for their valuable comments on the original manuscript.
Funders:
Funding AgencyGrant Number
NSFDMS-0353838
NSFACI-0204932
Subject Keywords:Euler Equation; Vortex Line; Vortex Tube; Vortex Sheet; Vorticity Vector
Issue or Number:6
Record Number:CaltechAUTHORS:20160322-090700507
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20160322-090700507
Official Citation:Hou, T. & Li, R. J Nonlinear Sci (2006) 16: 639. https://doi.org/10.1007/s00332-006-0800-3
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:65575
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:30 Mar 2016 23:51
Last Modified:15 Aug 2019 19:55

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