The interaction of skewed vortex pairs: a model for blow-up of the Navier–Stokes equations (original) (raw)

The interaction of two propagating vortex pairs is considered, each pair being initially aligned along the positive principal axis of strain associated with the other. As a preliminary, the action of accelerating strain on a Burgers vortex is considered and the conditions for a finite-time singularity (or ‘blow-up’) are determined. The asymptotic high Reynolds number behaviour of such a vortex under non-axisymmetric strain, and the corresponding behaviour of a vortex pair, are described. This leads naturally to consideration of the interaction of the two vortex pairs, and identifies a mechanism by which blow-up may occur through self-similar evolution in an interaction zone where scale decreases in proportion to (t* − t)1/2, where t* is the singularity time. The relevance of Leray scaling in this interaction zone is discussed.