Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: šil'nikov's chaos and the devil's staircase

Abstract

We study the motion of non-diffusive, passive particles within steady, three-dimensional vortex breakdown bubbles in a closed cylindrical container with a rotating bottom. The velocity fields are obtained by solving numerically the three-dimensional Navier–Stokes equations. We clarify the relationship between the manifold structure of axisymmetric (ideal) vortex breakdown bubbles and those of the three-dimensional real-life (laboratory) flow fields, which exhibit chaotic particle paths. We show that the upstream and downstream fixed hyperbolic points in the former are transformed into spiral-out and spiral-in saddles, respectively, in the latter. Material elements passing repeatedly through the two saddle foci undergo intense stretching and folding, leading to the growth of infinitely many Smale horseshoes and sensitive dependence on initial conditions via the mechanism discovered by šil'nikov (1965). Chaotic šil'nikov orbits spiral upward (from the spiral-in to the spiral-out saddle) around the axis and then downward near the surface, wrapping around the toroidal region in the interior of the bubble. Poincaré maps reveal that the dynamics of this region is rich and consistent with what we would generically anticipate for a mildly perturbed, volume-preserving, three-dimensional dynamical system (MacKay 1994; Mezić & Wiggins 1994a). Nested KAM-tori, cantori, and periodic islands are found embedded within stochastic regions. We calculate residence times of upstream-originating non-diffusive particles and show that when mapped to initial release locations the resulting maps exhibit fractal properties. We argue that there exists a Cantor set of initial conditions that leads to arbitrarily long residence times within the breakdown region. We also show that the emptying of the bubble does not take place in a continuous manner but rather in a sequence of discrete bursting events during which clusters of particles exit the bubble at once. A remarkable finding in this regard is that the rate at which an initial population of particles exits the breakdown region is described by the devil's staircase distribution, a fractal curve that has been already shown to describe a number of other chaotic physical systems.

Separatrix Splitting in 3D Volume-Preserving Maps

“…There exists also an one-dimensional straight heteroclinic connection between the fixed points. The same configuration appears in fluid dynamics under the name of Hill's spherical vortex or bubble-type vortex breakdown [32], and as a model for the magnetic field of stars, in which case it is called spheromak [25]. From a more theoretical point of view, we note that the integrable normal forms associated to families of volume-preserving flows with a Hopf-zero singularity have the same structure in the phase space [11].…”

“…The infinite dimensional group of volume-preserving diffeomorphisms on R is at the core of the ambitious program to reformulate hydrodynamics [3]. Volumepreserving maps arise in a number of applications such as the study of the motion of Lagrangian tracers in incompressible fluids or of the structure of magnetic field lines [18,19,32,29]. Experimental methods have only recently been developed that allow the visualization of particle trajectories in spatial fluids [27,31].…”

A coaxial vortex ring model for vortex breakdown

“…I } comprises a heteroclinic cycle that we denote as Z I , which is susceptible to perturbations that can generate chaotic dynamical regimes, such as described in [4,10,18,27,33,44,45,57,65].…”

Topology-guided Visualization of Constrained Vector Fields

“…10). We refer the reader to the paper of Sotiropoulos et al [SVL01] for details. In the field of visualization, vortex breakdown bubbles have been studied recently ([TGK*04, GTS04, GTS*04]).…”