Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1899851 | Physica D: Nonlinear Phenomena | 2009 | 12 Pages |
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−=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 αα near 3/2 and u+∼logξ, where ξ=0ξ=0 denotes the singularity surface. The logarithmic singular surface is related to the double exponential growth of vorticity observed in recent numerical simulations.