Numerical study of singularity formation in a class of Euler and Navier–Stokes flows

We study numerically a class of stretched solutions of the three-dimensional Euler and Navier–Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover,...

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Bibliographic Details
Published in:Physics of fluids (1994) 2000-12, Vol.12 (12), p.3181-3194
Main Authors: Ohkitani, Koji, Gibbon, John D.
Format: Article
Language:English
Subjects:
Axial flow
Navier Stokes equations
Numerical analysis
Velocity
Viscosity of liquids
Vortex flow
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Summary:We study numerically a class of stretched solutions of the three-dimensional Euler and Navier–Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier–Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the two-dimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.
ISSN:1070-6631
1089-7666
DOI:10.1063/1.1321256