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.

In a recent study, Sotiropoulos et al. (2001) studied numerically the chaotic particle paths in the interior of stationary vortex-breakdown bubbles that form in a closed cylindrical container with a rotating lid. Here we report the first experimental verification of these numerical findings along with new insights into the dynamics of vortex-breakdown bubbles. We visualize the Lagrangian transport within the bubbles using planar laser-induced fluorescence (LIF) and show that even though the flow fields are steady – from the Eulerian standpoint – the spatial distribution of the dye tracer varies continuously, and in a seemingly random manner, over very long observation intervals. This finding is consistent with the arbitrarily long šil'nikov transients of upstream-originating orbits documented numerically by Sotiropoulos et al. (2001). Sequences of instantaneous LIF images also show that the steady bubbles exchange fluid with the outer flow via random bursting events during which blobs of dye exit the bubble through the spiral-in saddle. We construct experimental Poincaré maps by time-averaging a sufficiently long sequence of instantaneous LIF images. Ergodic theory concepts (Mezić & Sotiropoulos 2002) can be used to formally show that the level sets of the resulting time-averaged light intensity field reveal the invariant sets (unmixed islands) of the flow. The experimental Poincaré maps are in good agreement with the numerical computations. We apply this method to visualize the dynamics in the interior of the vortex-breakdown bubble that forms in the wake of the first bubble for governing parameters in the steady, two-bubble regime. In striking contrast with the asymmetric image obtained for the first bubble, the time-averaged light intensity field for the second bubble is remarkably axisymmetric. Numerical computations confirm this finding and further reveal that the apparent axisymmetry of this bubble is due to the fact that orbits in its interior exhibit quasi-periodic dynamics. We argue that this stark contrast in dynamics should be attributed to differences in the swirl-to-axial velocity ratio in the vicinity of each bubble. By studying the bifurcations of a simple dynamical system, with manifold topology resembling that of a vortex-breakdown bubble, we show that sufficiently high swirl intensities can stabilize the chaotic orbits, leading to quasi-periodic dynamics.

We solve numerically the three-dimensional incompressible Navier-Stokes equations to simulate the flow in a cylindrical container of aspect ratio one with exactly counter-rotating lids for a range of Reynolds numbers for which the flow is steady and three dimensional (300⩽Re⩽850). In agreement with linear stability results [C. Nore et al., J. Fluid Mech. 511, 45 (2004)] we find steady, axisymmetric solutions for Re<300. For Re>300 the equatorial shear layer becomes unstable to steady azimuthal modes and a complex vortical flow emerges, which consists of cat’s eye radial vortices at the shear layer and azimuthally inclined axial vortices. Upon the onset of the three-dimensional instability the Lagrangian dynamics of the flow become chaotic. A striking finding of our work is that there is an optimal Reynolds number at which the stirring rate in the chaotically advected flow is maximized. Above this Reynolds number, the integrable (unmixed) part of the flow begins to grow and the stirring rate is shown conclusively to decline. This finding is explained in terms of and appears to support a recently proposed theory of chaotic advection [I. Mezić, J. Fluid Mech. 431, 347 (2001)]. Furthermore, the calculated rate of decay of the stirring rate with Reynolds numbers is consistent with the Re−1∕2 upper bound predicted by the theory.

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Model studies have been conducted to investigate the potential coral reef exposure from proposed dredging associated with development of a new deepwater wharf in outer Apra Harbor, Guam. The Particle Tracking Model (PTM) was applied to quantify the exposure of coral reefs to material suspended by the dredging operations at proposed sites. Key PTM features include the flexible capability of continuous multiple releases of sediment parcels, control of parcel/substrate interaction, and the ability to track vast numbers of parcels efficiently. This flexibility has allowed for model simulation of the combined effects of sediment release from clamshell dredging, of chiseling to fracture limestone blocks, of silt curtains, and of flocculation. Because the rate of material released into the water column by some of the processes is not well understood or a priori known, the modeling protocol was to bracket parameters within reasonable ranges to produce a suite of potential results from multiple model runs. Data analysis results include mapping the time histories and the maximum values of suspended sediment concentration and deposition rate. Following exposure modeling, the next phase of the analysis has been an ecological assessment to translate the PTM exposure level predictions into predicted amounts of coral reef damage. The level of potential coral reef impact will be an important component of the final selection process for the new deepwater berthing site.