Chaotic advection by two interacting finite-area vortices (original) (raw)
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An analytical study of transport, mixing and chaos in an unsteady vortical flow
Journal of Fluid Mechanics, 1990
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 field induced by a vortex pair plus an oscillating strainrate field. I n 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 neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour 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 flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. 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 required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this 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.
Chaotic transport and mixing of a passive admixture by vortex flows behind obstacles
Izvestiya, Atmospheric and Oceanic Physics, 2010
A two layer nonviscous model of chaotic advection in a unidirectional pulsating running current above a delta shaped underwater elevation is considered. The property of local stability is used and a charac teristic similar to the cumulative Lyapunov exponent is introduced that makes it possible to determine the range of regular and chaotic particle behavior. The estimates obtained using this characteristic are for clari fying passive admixture transport in analog model problems. Knowledge of the maximum chaotization region boundary is important for oceanology in view of interpreting point vortices as a model of distributed vortices. The criterion based on using the cubic Hamiltonian approximation for a nonlinear resonance model is intro duced to estimate the limiting boundary of the regular region.
Journal of Fluid Mechanics, 2003
The advection of a passive scalar blob in the deformation field of an axisymmetric vortex is a simple mixing protocol for which the advection-diffusion problem is amenable to a near-exact description. The blob rolls up in a spiral which ultimately fades away in the diluting medium. The complete transient concentration field in the spiral is accessible from the Fourier equations in a properly chosen frame. The concentration histogram of the scalar wrapped in the spiral presents unexpected singular transient features and its long time properties are discussed in connection with real mixtures.
Physical Review E, 2011
We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density n ≪ 1. When the more efficient three-body vortex mergings are considered (involving vortices of different signs), we show that n ∼ t −ξ , with ξ = 1. We generalize this argument to three-dimensional geostrophic turbulence, finding ξ = 5/4, in excellent agreement with direct Navier-Stokes simulations [
Dynamics AND Chaos IN Wavy Vortex Flow
Dynamics of transport and mixing in a Couette-Taylor device was investigated by computationally tracking fluid particles using a three-dimensional, three-component experimental velocity field determined with recent particle image velocimetry (PIV) measurements at conditions well above transition to wavy vortex flow. Results indicate enhanced mixing due to increased stretching and folding in all directions. This results in very effective axial dispersion of fluid particles. Waviness and increasing vortex strength results in enhanced axial dispersion as the Reynolds number increases.
Vortex-induced chaotic mixing in wavy channels
Journal of Fluid Mechanics, 2010
Mixing is studied in open-flow channels with conformally mapped wavy-wall profiles, using a point-vortex model in two-dimensional irrotational, incompressible mean flow. Unsteady dynamics of the separation bubble induced by oscillatory motion of point vortices located in the trough region produces chaotic mixing in the Lagrangian sense. Significant mass exchange between passive tracer particles inside and outside of the separation bubble forms an efficient mixing region which evolves in size as the vortex moves in the unsteady potential flow. The dynamics closely resembles that obtained by previous authors from numerical solutions of the unsteady Navier-Stokes equations for oscillatory unidirectional flow in a wavy channel. Of the wavy channels considered, the skew-symmetric form is most efficient at promoting passive mixing. Diffusion via gridless random walks increases lateral particle dispersion significantly at the expense of longitudinal particle dispersion due to the opposing effect of mass exchange at the front and rear of the particle ensemble. Active mixing in the wavy channel reveals that the fractal nature of the unstable manifold plays a crucial role in singular enhancement of productivity. Hyperbolic dynamics dominate over nonhyperbolicity which is restricted to the vortex core region. The model is simple yet qualitatively accurate, making it a potential candidate for the study of a wide range of vortex-induced transport and mixing problems.
Chaotic mixing by longitudinal vorticity
Chemical Engineering Science, 2013
Computational fluid dynamics study of the chaotic advection flow in static mixer. Mixing enhancement by using longitudinal vortex generator. Poincaré sections and the Lagrangian trajectories show that chaotic advection takes place in the alternated configuration.
Chaotic advection near a three-vortex collapse
Physical review. E, Statistical, nonlinear, and soft matter physics, 2001
Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system's initial conditions are set closer to those leading to the collapse of the vortices; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the core's radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a superdiffusive regime. The transport exponent varies with time decades, implying the presence of mu...
Chaotic advection of finite-size bodies in a cavity flow
Physics of Fluids, 2003
We considered advection of neutrally buoyant discs in two-dimensional chaotic Stokes flow. The goal of the study is to explore a possibility to enhance laminar mixing in batch-flow mixers. Addition of freely moving bodies to periodically driven chaotic flow renders the flowfield nonperiodic ͓D. F. Zhang and D. A. Zumbrunnen, AIChE J. 42, 3301 ͑1996͔͒, i.e., the Lagrangian chaos of the bodies motion induces Eulerian chaos of the flow that makes mixing more intensive. The presence of three bodies creates new topological features that do not exist in ''pure'' fluid. The trajectories of the discs in the augmented phase space tangle and form a braid that leads to so-called topological chaos ͓P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 ͑2000͔͒. Simulations were performed using a new variant of the immersed boundaries method that allows the direct numerical simulation of fluid-solid flows on a regular rectangular grid without explicit calculation of the forces that the particles exert on the fluid.
Passive scalar advection in the vicinity of two point vortices in a deformation flow
European Journal of Mechanics - B/Fluids, 2012
The dynamics of passive fluid particles in the vicinity of two point vortices with arbitrary intensities, embedded in a steady external deformation flow, is studied. The motion of passive fluid particles is described by a nonintegrable 1.5 degrees of freedom dynamical system. Though the external flow is stationary, the additional half degree of freedom appears because the vortices' motion about their stationary positions is periodic. Then, this periodic motion plays the role of a periodic perturbation for the system describing the passive particle dynamics. Therefore, chaotic advection of passive fluid particles in the vicinity of these two vortices can occur. If the vortices, however, are situated at their stationary positions, they become motionless, and the dynamical system describing the passive particles' dynamics is also stationary. In the case of motionless vortices, a classification of the phase portraits of the passive particle motion is conducted by analyzing the number of critical points. When the vortices do not lie at their stationary position, the system becomes nonstationary. In this case, the existence of impenetrable transport barriers for chaotic advection is shown. These barriers are destroyed when stochastic layers merge; these layers widen as the deviation of the vortex position from the stationary points, increases. The efficiency of chaotization is analyzed by means of Poincaré sections and accumulated Lyapunov exponents.