Calculation of complex singular solutions to the 3D incompressible Euler equations

Document Type

Article

Publication Date

1-1-2009

Abstract

This paper presents numerical computations of complex singular solutions to the 3D incompressible Euler equations. The Euler solutions found here consist of a complex valued velocity field u+ that contains all positive wavenumbers; u+ satisfies the usual Euler equations but with complex initial data. The real valued velocity u = u+ + u- (where u- = over(u, -)+) is an approximate singular solution to the Euler equations under Moore's approximation. The method for computing singular solutions is an extension of that in Caflisch (1993) [25] for axisymmetric flow with swirl, but with several improvements that prevent the extreme magnification of round-off error which affected previous computations. This enables the first clean analysis of the singular surface in three-dimensional complex space. We find singularities in the velocity field of the form u+ ∼ ξα - 1 for α near 3/2 and u+ ∼ log ξ, where ξ = 0 denotes the singularity surface. The logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations. © 2009 Elsevier B.V. All rights reserved.

Identifier

70350568782 (Scopus)

Publication Title

Physica D Nonlinear Phenomena

External Full Text Location

https://doi.org/10.1016/j.physd.2009.09.016

ISSN

01672789

First Page

2368

Last Page

2379

Issue

23-24

Volume

238

Grant

DMS-0420590

Fund Ref

National Science Foundation

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