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### 3D Euler about a 2D symmetry plane

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Bustamante, Miguel D. and Kerr, Robert M..
(2008)
*3D Euler about a 2D symmetry plane.*
Physica D: Nonlinear Phenomena, Vol.237
(No.14-17).
pp. 1912-1920.
ISSN 0167-2789

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1016/j.physd.2008.02.007

## Abstract

Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr [R.M. Kerr, Evidence for a singularity of the three-dimensional, incompressible Euler equations, Phys. Fluids 5 (1993) 1725-1746] and the lack thereof by Hou and Li [TY. Hou, R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci. 16 (2006) 639-664]. Core profiles designed to reproduce the two results are presented, new more robust analysis is proposed, and new criteria for when calculations should be terminated are introduced and compared with classical resolution studies and spectral convergence tests. Most of the analysis is on a 512 x 128 x 2048 mesh, with new analysis on a just completed 1024 x 256 x 2048 used to confirm trends. One might hypothesize that there is a finite-time singularity with enstrophy growth like Omega similar to (T-c - t)(-gamma Omega) and vorticity growth like parallel to omega parallel to(infinity) similar to (T-c - t)(-gamma). The new analysis would then support gamma Omega approximate to 1/2 and gamma > 1. These represent modifications of the conclusions of Kerr [op. cit.]. Issues that might arise at higher resolution are discussed. (C) 2008 Elsevier B.V. All rights reserved.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics Q Science > QC Physics |

Divisions: | Faculty of Science > Engineering Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Lagrange equations, Singularities (Mathematics), Fluid mechanics, Vortex-motion |

Journal or Publication Title: | Physica D: Nonlinear Phenomena |

Publisher: | Elsevier BV |

ISSN: | 0167-2789 |

Date: | 15 August 2008 |

Volume: | Vol.237 |

Number: | No.14-17 |

Number of Pages: | 9 |

Page Range: | pp. 1912-1920 |

Identification Number: | 10.1016/j.physd.2008.02.007 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Leverhulme Foundation, University of Warwick. Centre for Scientific Computing |

Grant number: | F/00 215/AC (LF) |

References: | [1] L. Euler, Principia motus fluidorum., Novi Commentarii Acad. Sci. Petropolitanae 6 (1761) 271–311. [2] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys. 94 (1984) 61. [3] R.M. Kerr, Evidence for a singularity of the three-dimensional, incompressible Euler equations, Phys. Fluids 5 (1993) 1725–1746. [4] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potentially singular solutions for the 3D Euler equations, Comm. Partial Differential Equations 21 (1996) 559–571. [5] R.M. Kerr, Velocity and scaling of collapsing Euler vortices, Phys. Fluids 17 (2005) 075103. [6] R. Grauer, C. Marliani, K. Germaschewski, Adaptive mesh refinement of singular solutions of the incompressible Euler equations, Phys. Rev. Lett. 80 (1998) 4177–4180. [7] P. Orlandi, G. Carnevale, Nonlinear amplification of vorticity in inviscid interaction of orthogonal Lamb dipoles, Phys. Fluids 19 (2007) 057106. [8] T.Y. Hou, R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci. 16 (2006) 639–664. [9] R.M. Kerr, F. Hussain, Simulation of vortex reconnection, Physica D 37 (1989) 474–484. [10] R.M. Kerr, Evidence for a singularity of the three-dimensional incompressible Euler equations, in: G.M. Zaslavsky, M. Tabor, P. Comte (Eds.), Topological Aspects of the Dynamics of Fluids and Plasmas, Proceedings of the NATO-ARW Workshop at the Institute for Theoretical Physics, University of California at Santa Barbara, Kluwer Academic Publishers, Dordrecht, Netherlands, 1992, pp. 309–336. [11] T.Y. Hou, R. Li, Computing nearly singular solutions using pseudospectral methods, J. Comp. Phys. 226 (2007) 379–397. [12] C. Sulem, P.-L. Sulem, H. Frisch, Tracing complex singularities with spectral methods, J. Comp. Phys. 50 (1983) 138–161. [13] C. Cichowlas, M.E. Brachet, Evolution of complex singularities in Kida- Pelz and Taylor-Green inviscid flows, Fluid Dynamics Res. 36 (2004) 239–248. [14] U. Frisch, T. Matsumoto, J. Bec, Singularities of Euler flow? Not out of the blue!, J. Stat. Phys. 113 (2003) 761–781. [15] J. Deng, T.Y. Hou, X. Yu, Improved geometric condition for non-blowup of the 3D incompressible Euler equation, Commun. Partial Differential Equations 31 (2006) 293–306. [16] T. Grafke, H. Homann, J. Dreher, R. Grauer, Numerical simulations of possible finite time singularities in the incompressible Euler equations: Comparison of numerical methods 2007 (these Proceedings). [17] M.E. Brachet, D.I. Meiron, S.A. Orszag, B.G. Nickel, R.H. Morf, U. Frisch, Small-scale structure of the Taylor–Green vortex, J. Fluid Mech. 130 (1983) 411–452. |

URI: | http://wrap.warwick.ac.uk/id/eprint/29497 |

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