scholarly journals Diffusion of a passive scalar from a no-slip boundary into a two-dimensional chaotic advection field

1998 ◽  
Vol 372 ◽  
pp. 119-163 ◽  
Author(s):  
S. GHOSH ◽  
A. LEONARD ◽  
S. WIGGINS
Keyword(s):  
Passive Scalar ◽  
Diffusion Problem ◽  
Chaotic Advection ◽  
Two Dimensional ◽  
Shadowing Property ◽  
Slip Boundary ◽  
Advection Diffusion ◽  
Scalar Diffusion ◽  
Time Periodic ◽  
Large Peclet Number

Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local ‘hot spot’ on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Péclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity field further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Péclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Péclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field.

Experiments on mixing due to chaotic advection in a cavity

1989 ◽  
Vol 209 ◽  
pp. 463-499 ◽  
Author(s):  
C. W. Leong ◽  
J. M. Ottino
Keyword(s):  
Large Scale ◽  
Spatial Arrangement ◽  
Periodic Points ◽  
Chaotic Advection ◽  
Two Dimensional ◽  
Complex Behaviour ◽  
Chaotic Flows ◽  
Regular Time ◽  
Slow Flows ◽  
Time Periodic

Chaotic mixing of fluids in slow flows is ubiquitous but incompletely understood. However, relatively simple experiments provide a wealth of information regarding mixing mechanisms and indicate the need for complementary theoretical developments in dynamical systems. In this work we presnt a versatile cavity flow apparatus, capable of producing a variety of two-dimensional velocity fields, and use it to conduct a detailed experimental study of mixing in low-Reynolds-number flows. Since the goal is detailed understanding, only two time-periodic co-rotating flows induced by wall motions are considered: one continuous and the other discontinuous. Both types of flows produce exponential growth of intermaterial area, as expected from chaotic flows, and a mixture of islands and chaotic regions. A procedure for identifying periodic points and determining their movements is presented as well as how to make meaningful comparisons between periodic flows. We observe that periodic points move very much as a planetary system; planets (hyperbolic points) have moons (elliptic points) with twice the period of the planets; furthermore the spatial arrangement of periodic points becomes symmetric at regular time intervals. Detailed analyses reveal complex behaviour: birth, bifurcation, and collapse of islands; formation and periodic motion of coherent structures, such as islands and large-scale folds. However, the richness and complexity of the results obtained indicate that these two-dimensional time-periodic systems are far from completely understood and that other wall motions might deserve a similar level of scrutiny.

scholarly journals Coherent structures and chaotic advection in three dimensions

2010 ◽  
Vol 654 ◽  
pp. 1-4 ◽  
Author(s):  
STEPHEN WIGGINS
Keyword(s):  
Dynamical Systems ◽  
Fluid Mechanics ◽  
Coherent Structures ◽  
Three Dimensional ◽  
Point Of View ◽  
Three Dimensions ◽  
Chaotic Advection ◽  
Periodic Flow ◽  
Two Dimensional ◽  
Time Periodic

In the 1980s the incorporation of ideas from dynamical systems theory into theoretical fluid mechanics, reinforced by elegant experiments, fundamentally changed the way in which we view and analyse Lagrangian transport. The majority of work along these lines was restricted to two-dimensional flows and the generalization of the dynamical systems point of view to fully three-dimensional flows has seen less progress. This situation may now change with the work of Pouransari et al. (J. Fluid Mech., this issue, vol. 654, 2010, pp. 5–34) who study transport in a three-dimensional time-periodic flow and show that completely new types of dynamical systems structures and consequently, coherent structures, form a geometrical template governing transport.

Mixing in frozen and time-periodic two-dimensional vortical flows

2001 ◽  
Vol 442 ◽  
pp. 359-385 ◽  
Author(s):  
ALEXANDER WONHAS ◽  
J. C. VASSILICOS
Keyword(s):  
Scalar Field ◽  
Passive Scalar ◽  
Time Range ◽  
Two Dimensional ◽  
Kam Tori ◽  
Advection Diffusion ◽  
Box Counting Dimension ◽  
Theoretical Predictions ◽  
Scalar Variance ◽  
Variance Decay

In the first part of this paper, we investigate passive scalar or tracer advection–diffusion in frozen, two-dimensional, non-circular symmetric vortices. We develop an asymptotic description of the scalar field in a time range 1 [Lt ] t/T [Lt ] Pe1/3, where T is the formation time of the spiral in the vortex and Pe is a Péclet number, assumed much larger than 1. We derive the leading-order decay of the scalar variance E(t) for a singular non-circular streamline geometry,The variance decay is solely determined by a geometrical parameter μ and the exponent β describing the behaviour of the closed streamline periods. We develop a method to predict, in principle, the variance decay from snapshots of the advected scalar field by reconstructing the streamlines and their period from just two snapshots of the advected scalar field.In the second part of the paper, we investigate variance decay in a periodically moving singular vortex. We identify three different regions (core, chaotic and KAM-tori). We find fast mixing in the chaotic region and investigate a conjecture about mixing in the KAM-tori region. The conjecture enables us to use the results from the first section and relates the Kolmogorov capacity, or box-counting dimension, of the advected scalar to the decay of the scalar variance. We check our theoretical predictions against a numerical simulation of advection–diffusion of scalar in such a flow.

FRACTIONAL STEP METHODS FOR SPECTRAL APPROXIMATION OF ADVECTION-DIFFUSION EQUATIONS

1996 ◽  
Vol 06 (07) ◽  
pp. 1027-1050 ◽  
Author(s):  
PAOLA GERVASIO
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Peclet Number ◽  
Diffusion Equations ◽  
Fractional Step ◽  
Spectral Approximation ◽  
Two Dimensional ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Large Peclet Number

Several classical fractional step schemes are proposed for the spectral approximation of advection-diffusion equation in two-dimensional geometries. Suitable boundary conditions are studied in order to preserve the accuracy of the schemes at each step. An example is given for the application of the fractional step schemes to solve problems with large Péclet number.

How vortices mix

2003 ◽  
Vol 476 ◽  
pp. 213-222 ◽  
Author(s):  
P. MEUNIER ◽  
E. VILLERMAUX
Keyword(s):  
Concentration Field ◽  
Passive Scalar ◽  
Diffusion Problem ◽  
Deformation Field ◽  
Advection Diffusion ◽  
Axisymmetric Vortex ◽  
Long Time ◽  
Exact Description

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.

scholarly journals Unsteady feeding and optimal strokes of model ciliates

2013 ◽  
Vol 715 ◽  
pp. 1-31 ◽  
Author(s):  
Sébastien Michelin ◽  
Eric Lauga
Keyword(s):  
Peclet Number ◽  
Feeding Rate ◽  
Diffusion Problem ◽  
Advective Transport ◽  
Péclet Number ◽  
Fixed Energy ◽  
Advection Diffusion ◽  
The Mean ◽  
Micro Organisms ◽  
Time Periodic

AbstractThe flow field created by swimming micro-organisms not only enables their locomotion but also leads to advective transport of nutrients. In this paper we address analytically and computationally the link between unsteady feeding and unsteady swimming on a model micro-organism, the spherical squirmer, actuating the fluid in a time-periodic manner. We start by performing asymptotic calculations at low Péclet number ($\mathit{Pe}$) on the advection–diffusion problem for the nutrients. We show that the mean rate of feeding as well as its fluctuations in time depend only on the swimming modes of the squirmer up to order ${\mathit{Pe}}^{3/ 2} $, even when no swimming occurs on average, while the influence of non-swimming modes comes in only at order ${\mathit{Pe}}^{2} $. We also show that generically we expect a phase delay between feeding and swimming of $1/ 8\mathrm{th} $ of a period. Numerical computations for illustrative strokes at finite $\mathit{Pe}$ confirm quantitatively our analytical results linking swimming and feeding. We finally derive, and use, an adjoint-based optimization algorithm to determine the optimal unsteady strokes maximizing feeding rate for a fixed energy budget. The overall optimal feeder is always the optimal steady swimmer. Within the set of time-periodic strokes, the optimal feeding strokes are found to be equivalent to those optimizing periodic swimming for all values of the Péclet number, and correspond to a regularization of the overall steady optimal.

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