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Published 2008 | Submitted + Published
Book Section - Chapter Open

Numerical Study of Nearly Singular Solutions of the 3-D Incompressible Euler Equations


In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously investigated by Kerr in [16, 19]. In our numerical study, we use both the pseudo-spectral method with the 2/3 dealiasing rule and the pseudo-spectral method with a high order Fourier smoothing. Moreover, we perform a careful resolution study with grid points as large as 1,536 × 1,024 × 3,072 to demonstrate the convergence of both numerical methods. Our computational results show that the maximum vorticity does not grow faster than doubly exponential in time while the velocity field remains bounded up to T = 19, beyond the singularity time T = 18.7 reported by Kerr in [16, 19]. The local geometric regularity of vortex lines near the region of maximum vorticity seems to play an important role in depleting the nonlinear vortex stretching dynamically.

Additional Information

© 2008 Springer-Verlag Berlin Heidelberg. We would like to thank Prof. Lin-Bo Zhang from the Institute of Computational Mathematics in Chinese Academy of Sciences for providing us with the computing resource to perform this large scale computational project. Additional computing resource was provided by the Center of Super Computing Center of Chinese Academy of Sciences. We also thank Prof. Robert Kerr for providing us with his Fortran subroutine that generates his initial data. This work was in part supported by NSF under the NSF FRG grant DMS-0353838 and DMS-0713670. 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.

Attached Files

Submitted - 0608126.pdf

Published - Hou-Li2008_Chapter_NumericalStudyOfNearlySingular.pdf


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August 19, 2023
August 19, 2023