Part I. An analytical study of transport, mixing and chaos in an unsteady vortical flow. Part II. Transport in two dimensional maps

Abstract

PART I: We examine the transport properties of a particular two dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the velocity field induced by a vortex pair plus an oscillating strain-rate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighborhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomena and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical study includes the interpretation of the invariant manifolds as the underlying structure which govern the transport. Then we use Melnikov's technique to investigate the behavior of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flows. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighborhood. We develop a technique for determining the residence time distribution for fluid particles near the vortices. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort for the residence time distributions. Additionally, we develop a finite time analog of the Liapunov exponent which measures the effect of, the horseshoes on trajectories passing through the mixing region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment, and the flux rate. PART II: We study transport in the two dimensional phase space of Cr diffeomorphisms (r >= 1) of two manifolds between regions of the phase space bounded by pieces of the stable and unstable manifolds of hyperbolic fixed points. The mechanism for the transport is associated with the dynamics of homoclinic and heteroclinic tangles and the study of this dynamics leads to a general formulation for the transport rates in terms of distributions of small regions in phase space (lobes). It is shown how the method applies to three geometrical configurations, one of which corresponds to the geometry associated with the Kelvin-Stuart Cat's eye flow undergoing a time periodic perturbation. In this case the formulae imply, for example, that the evolution of only two lobes determines the mass transport from the upper to the lower half plane of the fluid flow. As opposed to previous studies this formulation takes into account the effect of re-entrainment of the lobes, i.e. the implications of the lobes leaving and re-entering the specified regions on the transport rates. The formulation is developed for both area preserving and non area preserving two dimensional diffeomorphisms and does not require the map to be near integrable. The techniques involved in applying this formulation are discussed including the possible use of the generating function for computing the distributions of the lobes in phase space, and the use of Poincare maps which enables one to study the transport in continuous time systems via the above formalism. In particular, we demonstrate how the right choice of the Poincare section can reduce the calculations of the transport rates.