Evolution and breakdown of a vortex street in two dimensions

Abstract

The initial-value problem defined by two parallel vortex sheets of opposite sign is studied. Strictly two-dimensional, incompressible, nearly inviscid dynamics is assumed throughout. The roll-up of the sheets into a vortex street is simulated numerically using 4096 point vortices. Much longer runs than in previous work are performed, and it is found that only for a finite range of values of the ratio,h/λ, of sheet separation to perturbation wavelength, does a long-lived vortex street emerge. Forh/λ [gsim ] 0·6 a pairing transition within each row intervenes. Forh/λ [lsim ] 0·3 we find oscillatory modes.Using up to 16384 point vortices, we also study the breakdown of the metastable street to a two-dimensional, turbulent shear flow. The vortex blobs that made up the street may merge with others of the same sign after the breakdown, but otherwise persist throughout the turbulent regime. Neither their disintegration nor amalgamation with vortices of opposite sign was observed. Using dimensional arguments we derive the relevant scaling theory, and show that it applies to a flow started from two random vortex sheets. The resulting turbulence is not self-similar. For the turbulent flow that follows from the breakdown of a regular vortex street two length scales with different power-law growth in time appear to be necessary. The important differences in the asymptotic structure of flows initialized from random and regular sheets leads us to question the idea of universality. The influence of the symmetry of the initial perturbation on the subsequent development is also considered.