Advection-diffusion of a passive scalar in the flow of a decaying vortex

2005 ◽  
pp. 31-36 ◽  
Author(s):  
Konrad Bajer ◽  
Andrew P. Bassom ◽  
Andrew D. Gilbert
Keyword(s):  
Passive Scalar ◽  
Advection Diffusion

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.

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.

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 Simple passive scalar advection-diffusion model

1998 ◽  
Vol 58 (5) ◽  
pp. 5757-5764 ◽  
Author(s):  
Scott Wunsch
Keyword(s):  
Diffusion Model ◽  
Passive Scalar ◽  
Advection Diffusion

Large magnetic Reynolds number dynamo action in a spatially periodic flow with mean motion

1990 ◽  
Vol 331 (1621) ◽  
pp. 649-733 ◽  
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Asymptotic Theory ◽  
Passive Scalar ◽  
Magnetic Reynolds Number ◽  
Stagnation Points ◽  
Advection Diffusion ◽  
Asymptotic Expressions ◽  
Spatially Periodic ◽  
And Diffusion

We consider a problem of advection and diffusion of passive scalar and vector fields in a particular family of steady fluid flows. These flows are obtained by adding a small uniform velocity to a spatially periodic array of spiral eddies. The uniform flow, i/H, is taken to have the discrete form aH = e(M ,N ,0 )/(M 2 + N 2 )K 1, where M, N are relatively prime integers. The spatially periodic part, u may be expressed in terms of a streamfunction x]r', u' = (.........) = sin x sin y where K is a constant. The flow we study is therefore u = wH + w/. Our work is motivated by applications of dynamo theory and to classical diffusion of passive scalars. The above family of flows was chosen as typical of spatially periodic flow with non-zero mean velocity, ffH. The flows are comparatively simple because they are independent of z. Nevertheless the projection of the streamline pattern onto the plane z = 0 can be surprisingly complex, owing to the structure of u modulo the cell of periodicity of u'. This structure accounts for our special form of SH above, which makes the tangent of the angle of inclination of the uniform current a rational number. This uniform component breaks up the eddy pattern into closed eddies whose bounding streamlines begin and end at X-type stagnation points. The set of all such streamlines define the boundaries of the open channels, which fill the regions between the closed eddies and lie near the separatrices of u’. Then, for example, when is even the channel structure repeats under a shift ( , in the xy plane, leading to a periodicity in channel length of order L. Analogous results apply to the case L odd. This geometry raises interesting questions regarding the advection and diffusion of fields in the irrational limit, i.e. when M, N->■ oo, -*irrational. A basic result of this paper will be formal asymptotic expressions, for average physical quantities of interest, in the irrational limit. An asymptotic theory of advection-diffusion is exploited, based upon a separation into closed eddies, channels, and separatrix boundary layers. The fundamental assumption is that the dimensionless parameter R (a magnetic Reynolds number in the dynamo problem, a Peclet number in diffusion problems) is large, meaning that transport by molecular diffusion is nominally small compared with transport by advection. For large R, the X-type stagnation points trigger boundary layers, which for given MN, extend a distance of order L before repeating the structure. This leads to channel boundary layers of width L*R~*, compared with eddy boundary layer of width R~5 and channel widths of order eZT1, the eddies being separated by gaps of widths order e. In this setting the irrational limit is taken after the above asymptotic structure is isolated by the limit R-> oo. Our results consist of numerical studies for = eR? of order unity, and analytic asymptotic expressions derived under the condition > D. In the former, the eddy separation width is comparable with the eddy boundary layer width, so that we study the transition from transport dominated by boundary layers to transport dominated by channels. In the asymptotic theory for large the boundary-layer contributions may be neglected and the problem reduces to the analysis of channel geometry. Even here, the condition ft> Drestricts us to a countable set of mean flow orientations. The relation between solutions for these special orientations, and their immediate neighbours with irrational tangents, is discussed. Representative results for effective diffusion of a passive scalar field, and for mean induced electromotive force in an electrically conducting fluid (the a-effect) are presented. We also discuss the present examples in relation to the more complex problem of advection—diffusion by flows with chaotic lagrangian paths.

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