The kinematics and dynamics of vorticity in the inertial range: a possible scenario

The evolution of initially weak structures of vorticity as they evolve in an incompressible turbulent flow is investigated. Such objects are candidates for being important structures in the inertial range and in the dissipation range of scales. Initially, these structures evolve passively as a result of the induced velocity field of the large-scale vorticity field. This field is three dimensional and time dependent, so these objects are subjected to straining apropos of Lagrangian chaos, characterized by a distribution of finite-time Lyapunov exponents

Face-centred cubic lattices and particle redistribution in vortex methods

In vortex particle methods one is concerned with the problem of clustering and depletion of particles in different regions of the flow. The overlap of the vortex blobs is indeed of primary importance for the convergence of the method. In this paper we consider face-centred cubic (FCC) lattices for particle redistribution in three dimensions. This lattice is in fact the most natural way to pack spheres (the FCC is also known as a closest-sphere packing lattice). As a consequence, a point has 12 equidistant close neighbours rather than six for the cubic lattice. The FCC lattice thus offers some symmetry properties that should prove useful for a number of reasons, e.g., the core overlap issue. A few results for this scheme are presented. The problem of two colliding vortex rings at Re = 250 and 500 is studied with both the FCC and cubic lattice schemes. This problem subjects the vortex tubes to a quite strong stretching field and can amply test the quality of the lattice and the remeshing

Quantized turbulence physics

We elaborate the physics of systems of unconstrained, reconnecting vortex filaments with dynamic finite cores of uniform ("quantized") circulation interacting via Biot-Savart and viscous forces. The phenomenology of this purely structured turbulent system includes an inertial range with Kolmogorov's k^–5/3 scaling for the energy spectrum, as well as Kolmogorov's linear in r scaling for the third order longitudinal structure function

Velocity autocorrelations of decaying isotropic homogeneous turbulence

Velocity autocorrelations and the mean-square displacements of fluid particles are obtained for decaying, isotropic homogeneous turbulence by numerical simulation of the flow field, using 1283 and 2563 grids, and tracking several tens of thousands of fluid particles, using a third-order interpolation scheme. A self-preserving Lagrangian velocity autocorrelation coefficient is found in terms of a dimensionless time variable s, defined by ds=dt/[script T]s(t), under the observation of a power-law energy decay and the assumption that [script T]s(t) is proportional to the Lagrangian integral timescale [script T][script L]. This timescale is in turn assumed to be proportional to the length scale of the energy-containing eddies [script L]e~K3/2/epsilon divided by the turbulent velocity u[prime], where K=3/2u[prime]2 is turbulent energy and epsilon is the energy dissipation rate

A global study of enhanced stretching and diffusion in chaotic tangles

A global, finite-time study is made of stretching and diffusion in a class of chaotic tangles associated with fluids described by periodically forced two-dimensional dynamical systems. Invariant lobe structures formed by intersecting global stable and unstable manifolds of persisting invariant hyperbolic sets provide the geometrical framework for studying stretching of interfaces and diffusion of passive scalars across these interfaces. In particular, the present study focuses on the material curve that initially lies on the unstable manifold segment of the boundary of the entraining turnstile lobe.A knowledge of the stretch profile of a corresponding curve that evolves according to the unperturbed flow, coupled with an appreciation of a symbolic dynamics that applies to the entire original material curve in the perturbed flow, provides the framework for understanding the mechanism for, and topology of, enhanced stretching in chaotic tangles. Secondary intersection points (SIP's) of the stable and unstable manifolds are particularly relevant to the topology, and the perturbed stretch profile is understood in terms of the unperturbed stretch profile approximately repeating itself on smaller and smaller scales. For sufficiently thin diffusion zones, diffusion of passive scalars across interfaces can be treated as a one-dimensional process, and diffusion rates across interfaces are directly related to the stretch history of the interface.An understanding of interface stretching thus directly translates to an understanding of diffusion across interfaces. However, a notable exception to the thin diffusion zone approximation occurs when an interface folds on top of itself so that neighboring diffusion zones overlap. An analysis which takes into account the overlap of nearest neighbor diffusion zones is presented, which is sufficient to capture new phenomena relevant to efficiency of mixing. The analysis adds to the concentration profile a saturation term that depends on the distance between neighboring segments of the interface. Efficiency of diffusion thus depends not only on efficiency of stretching along the interface, but on how this stretching is distributed relative to the distance between neighboring segments of the interface

Direct numerical simulations of vortex rings at ReΓ = 7500

We present direct numerical simulations of the turbulent decay of vortex rings with ReΓ = 7500. We analyse the vortex dynamics during the nonlinear stage of the instability along with the structure of the vortex wake during the turbulent stage. These simulations enable the quantification of vorticity dynamics and their correlation with structures from dye visualization and the observations of circulation decay that have been reported in related experimental works. Movies are available with the online version of the paper

Reconnection of Colliding Vortex Rings

We investigate numerically the Navier-Stokes dynamics of reconnecting vortex rings at small Reynolds number for a variety of configurations. We find that reconnections are dissipative due to the smoothing of vorticity gradients at reconnection kinks and to the formation of secondary structures of stretched antiparallel vorticity which transfer kinetic energy to small scales where it is subsequently dissipated efficiently. In addition, the relaxation of the reconnection kinks excites Kelvin waves which due to strong damping are of low wave number and affect directly only large scale properties of the flow

Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems

The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles

Dynamical systems analysis of fluid transport in time-periodic vortex ring flows

It is known that the stable and unstable manifolds of dynamical systems theory provide a powerful tool for understanding Lagrangian aspects of time-periodic flows. In this work we consider two time-periodic vortex ring flows. The first is a vortex ring with an elliptical core. The manifolds provide information about entrainment and detrainment of irrotational fluid into and out of the volume transported with the ring. The likeness of the manifolds with features observed in flow visualization experiments of turbulent vortex rings suggests that a similar process might be at play. However, what precise modes of unsteadiness are responsible for stirring in a turbulent vortex ring is left as an open question. The second situation is that of two leapfrogging rings. The unstable manifold shows striking agreement with even the fine features of smoke visualization photographs, suggesting that fluid elements in the vicinity of the manifold are drawn out along it and begin to reveal its structure. We suggest that interpretations of these photographs that argue for complex vorticity dynamics ought to be reconsidered. Recently, theoretical and computational tools have been developed to locate structures analogous to stable and unstable manifolds in aperiodic, or finite-time systems. The usefulness of these analogs is demonstrated, using vortex ring flows as an example, in the paper by Shadden, Dabiri, and Marsden [Phys. Fluids 18, 047105 (2006)]