Anomalous diffusion and anomalous stretching in vortical flows

Physica D: Nonlinear Phenomena, 1994

Long-term particle tracking is used to study chaotic transport experimentally in laminar, chaotic, and turbulent flows in an annular tank that rotates sufficiently rapidly to insure two-dimensionality of the flow. For the laminar and chaotic velocity fields, the flow consists of a chain of vortices sandwiched between unbounded jets. In these flow regimes, tracer particles stick for long times to remnants of invariant surfaces around the vortices, then make long excursions ("flights") in the jet regions. The probability distributions for the flight time durations exhibit power-law rather than exponential decays, indicating that the particle trajectories are described mathematically as L6vy flights (i.e. the trajectories have infinite mean square displacement per flight). Sticking time probability distributions are also characterized by power laws, as found in previous numerical studies. The mixing of an ensemble of tracer particles is superdiffusive: the variance of the displacement grows with time as t * with 1 < 3' < 2. The dependence of the diffusion exponent 3' and the scaling of the probability distributions are investigated for periodic and chaotic flow regimes, and the results are found to be consistent with theoretical predictions relating L6vy flights and anomalous diffusion. For a turbulent flow, the L6vy flight description no longer applies, and mixing no longer appears superdiffusive.

Annals of the New York Academy of Sciences, 1993

Doklady Physics, 2010

Physics of Fluids, 2001

This article deals with the advection of fluid particles in the velocity field of two identical vortices with various vorticity distributions. The two-dimensional velocity field is aperiodic in the range of parameters studied here, namely, the neighborhood of the critical distance for merger. Ideas and methods from the theory of transport in dynamical systems are used to describe and quantify particle advection. These methods are applied to the numerical representation of the velocity field, which is obtained by solving the Euler equations with the vortex-in-cell method. It is found that the strongest stirring of vortex fluid occurs slightly above the critical distance for merger. In this regime the fluid located between the vortices is subjected to intense stirring, and some vortex fluid may be entrained into the chaotic region depending on the smoothness of the vorticity distribution. Initial conditions below the critical distance lead to stirring of fluid mainly before merger. In this case the flow geometry is used to quantify the efficiency of merger, which is defined as the ratio of the circulation of the resultant vortex to the total circulation of the original vortices. It is found that the vortices with the smoothest vorticity profile have the lowest efficiency. Experimental visualizations in a two-dimensional rotating fluid confirm the intense stretching and folding of fluid elements that occurs before the vortices merge.

Reviews of Modern Physics

This work reviews the present position of and surveys future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows. Contents 3 3. Mixing in microdroplets C. Biology 51 1. Ciliary and flagellar chaotic advection 51 2. Biological activity in chaotic oceanic flows 53 IX. Perspectives 55 A. How to choose the best stirring protocol for a given mixing problem? 56 B. Effect of fluid inertia 57 C. How to control mixing? 58 D. Dynamics of the wall 59 E. Strange eigenmodes 60 F. 3D unsteady flows 60 G. Synthesis 61 Acknowledgments 62 References 62

Fluid Dynamics Research, 2008

The dimensionless effective axial diffusion coefficient, D z , calculated from particle trajectories in steady wavy vortex flow in a narrow gap Taylor-Couette system, has been determined as a function of Reynolds number (R = Re/Re c ), axial wavelength ( z ), and the number of azimuthal waves (m). Two regimes of Reynolds number were found: (i) when R < 3.5, D z has a complex and sometimes multi-modal dependence on Reynolds number; (ii) when R > 3.5, D z decreases monotonically.

Physical Review E, 2008

Particle sedimentation in the vicinity of a fixed horizontal vortex with time-dependent intensity can be chaotic, provided gravity is sufficient to displace the particle cloud whilst the vortex is off or weak. This "stretch, sediment & fold" mechanism is close to the so-called blinking vortex effect, which is responsible for chaotic transport of perfect tracers, except that in the present case the vortex motion is replaced by gravitational settling. In the present work this phenomenon is analyzed for heavy Stokes particles moving under the sole effect of gravity and of a linear drag. The vortex is taken to be a fixed isolated point vortex the intensity of which varies under the effect of either boundary conditions or volume force. When the unsteadiness of the vortex is weak and the free-fall velocity is of the order of the fluid velocity, and the particle response time is small, the particle motion equation can be written asymptotically as a perturbed hamiltonian system the phase portrait of which displays a homoclinic trajectory. A homoclinic bifurcation is therefore likely to occur, and the contribution of particle inertia to the occurrence of this bifurcation is analyzed asymptotically by using Melnikov's method.

… Analysis and Prediction …

As more high-resolution observations become available, our view of ocean mesoscale turbulence more closely becomes that of a "sea of eddies." The presence of the coherent vortices significantly affects the dynamics and the statistical properties of mesoscale flows, with important consequences on tracer dispersion and ocean stirring and mixing processes. Here we review some of the properties of particle transport in vortex-dominated flows, concentrating on the statistical properties induced by the presence of an ensemble of vortices. We discuss a possible parameterization of particle dispersion in vortex-dominated flows, adopting the view that ocean mesoscale turbulence is a two-component fluid which includes intense, localized vortical structures with non-local effects immersed in a Kolmogorovian, low-energy turbulent background which has mostly local effects. Finally, we report on some recent results regarding the role of coherent mesoscale eddies in marine ecosystem functioning, which is related to the effects that vortices have on nutrient supply. frequency mesoscale variability (i.e. the medium-size fluctuations in the general circulation), for the mesoscale and sub-mesoscale coherent vortices (vortical motions at scales smaller than the internal Rossby radius of deformation, McWilliams 1985), or for a generic complicated motion in the presence of turbulence.

Birkhäuser Basel eBooks, 1999

Journal of Fluid Mechanics, 1986

The particle paths of the Arnold-Beltrami-Childress (ABC) flows u = (A sinz+C cosy, B sinz+A cosz, C siny+B cosx). are investigated both analytically and numerically. This three-parameter family of spatially periodic flows provides a simple steady-state solution of Euler's equations. Nevertheless, the streamlines have a complicated Lagrangian structure which is studied here with dynamical systems tools. In general, there is a set of closed (on the torus, T3) helical streamlines, each of which is surrounded by a finite region of KAM invariant surfaces. For certain values of the parameters strong resonances occur which disrupt the surfaces. The remaining space is occupied by chaotic particle paths: here stagnation points may occur and, when they do, they are connected by a web of heteroclinic streamlines. When one of the parameters A, B or C vanishes the flow is integrable. In the neighbourhood, perturbation techniques can be used to predict strong resonances. A systematic search for integrable cases is done using Painlev6 tests, i.e. studying complex-time singularities of fluid-particle trajectories. When ABC =k 0 recursive clustering of complex time singularities occurs that seems characteristic of nonintegrable behaviour.