Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

Camassa, Roberto; Martinsen-Burrell, Neil; McLaughlin, Richard M.

doi:10.1063/1.2778451

Cited by 6 publications

(10 citation statements)

References 47 publications

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“…As ⑀ → 0, we expect to find a class of modes in which the ground-state element reproduces the long-lived structures observed in a periodic domain of O͑1͒ period, at large Pe by Camassa et al 9 We employ a different approach from the regular perturbation expansion adopted for the Taylor modes. Asymptotics are obtained via WKBJ method.…”

Section: B the Limit ⑀ \ 0: Anomalous Modesmentioning

confidence: 93%

“…The coupling between convection and bare molecular diffusion in such setups can still result in overall anomalous diffusion, but distinguished from the classical Taylor regime due to the existence of long-lived modes 9 ͑see also Ref. 10 for scaling arguments͒.…”

Section: Introductionmentioning

confidence: 97%

“…10 for scaling arguments͒. Camassa et al 9 have recently considered this category of flows in a study aimed at investigating the effect of shear on the statistical evolution of a random, Gaussian, and small scale distribution of dye. They observed that the probability distribution function ͑PDF͒ migrates toward an intermittent regime ͑stretched-exponential tailed PDF͒.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of passive scalar advection in parallel shear flows: Sorting of modes at intermediate time scales

2010

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The time evolution of a passive scalar advected by parallel shear flows is studied for a class of rapidly varying initial data. Such situations are of practical importance in a wide range of applications from microfluidics to geophysics. In these contexts, it is well-known that the long-time evolution of the tracer concentration is governed by Taylor's asymptotic theory of dispersion. In contrast, we focus here on the evolution of the tracer at intermediate time scales. We show how intermediate regimes can be identified before Taylor's, and in particular, how the Taylor regime can be delayed indefinitely by properly manufactured initial data. A complete characterization of the sorting of these time scales and their associated spatial structures is presented. These analytical predictions are compared with highly resolved numerical simulations. Specifically, this comparison is carried out for the case of periodic variations in the streamwise direction on the short scale with envelope modulations on the long scales, and show how this structure can lead to "anomalously" diffusive transients in the evolution of the scalar onto the ultimate regime governed by Taylor dispersion. Mathematically, the occurrence of these transients can be viewed as a competition in the asymptotic dominance between large Péclet ͑Pe͒ numbers and the long/short scale aspect ratios ͑L Vel / L Tracer ϵ k͒, two independent nondimensional parameters of the problem. We provide analytical predictions of the associated time scales by a modal analysis of the eigenvalue problem arising in the separation of variables of the governing advection-diffusion equation. The anomalous time scale in the asymptotic limit of large k Pe is derived for the short scale periodic structure of the scalar's initial data, for both exactly solvable cases and in general with WKBJ analysis. In particular, the exactly solvable sawtooth flow is especially important in that it provides a short cut to the exact solution to the eigenvalue problem for the physically relevant vanishing Neumann boundary conditions in linear-shear channel flow. We show that the life of the corresponding modes at large Pe for this case is shorter than the ones arising from shear free zones in the fluid's interior. A WKBJ study of the latter modes provides a longer intermediate time evolution. This part of the analysis is technical, as the corresponding spectrum is dominated by asymptotically coalescing turning points in the limit of large Pe numbers. When large scale initial data components are present, the transient regime of the WKBJ ͑anomalous͒ modes evolves into one governed by Taylor dispersion. This is studied by a regular perturbation expansion of the spectrum in the small wavenumber regimes.

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Section: B the Limit ⑀ \ 0: Anomalous Modesmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Analysis of passive scalar advection in parallel shear flows: Sorting of modes at intermediate time scales

2010

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“…While such fields are inherently integrable, and hence explicitly non-chaotic, the results presented are of interest in the study of 'strange eigenmodes' of the advection diffusion equation in the Batchelor regime. First, the emergence of strange eigenmodes is independent of the integrability or non-integrability of the underlying flow [5], and indeed, the simple example considered here produces nontrivial periodic patterns. Second, the results shown for even these extremely simple cases point to the delicate relationship between the non-linearity of the conservative advection operator and small diffusivity.…”

Section: Discussionmentioning

confidence: 98%

Averaged dynamics of time-periodic advection diffusion equations in the limit of small diffusivity

Schäfer

Poje

Vukadinovic

2009

Physica D: Nonlinear Phenomena

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a b s t r a c tWe study the effect of advection and small diffusion on passive tracers. The advecting velocity field is assumed to have mean zero and to possess time-periodic stream lines. Using a canonical transform to action-angle variables followed by a Lie transform, we derive an averaged equation describing the effective motion of the tracers. An estimate for the time validity of the first-order approximation is established. For particular cases of a regularized vortical flow we present explicit formulas for the coefficients of the averaged equation both at first and at second order. Numerical simulations indicate that the validity of the above first-order estimate extends to the second order.

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“…By suitable combinations of Fourier modes we can easily write down flows that have zero normal flow n · u = 0 or obey no-slip u = 0 on the boundary C of D. We then simulate passive scalars in these flows using a code which is completely spectral. We note that several other studies have used flows with 2π-periodicity, but with a focus on the presence of islands and other structures in the fluid flow, rather than on boundary conditions for the fluid flow and scalar (Cerbelli, Adrover & Giona, 2003;Giona et al, 2004;Camassa, Martinsen-Burrell & McLaughlin, 2007;Turner, Thuburn & Gilbert, 2009). Crucial in our study is the use of symmetries: if the flow and initial condition are chosen suitably then evolution under the advection-diffusion equation preserves the Dirichlet or Neumann boundary conditions exactly for the case of slip flow, or approximately in the case of no-slip flow.…”

Section: Introductionmentioning

confidence: 99%

Passive scalar decay in chaotic flows with boundaries

Zaggout¹,

Gilbert²

2012

Fluid Dyn. Res.

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This paper considers the long-time decay rate of a passive scalar in two-dimensional flow. The focus is on the effects of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain D. The boundary conditions on the passive scalar are imposed on the boundary of D by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case.At late times the decay of a passive scalar is exponential in time with a decay rate γ(κ), where κ is the molecular diffusivity. Scaling laws of the form γ(κ) Cκ α for small κ are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, α 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that α 1/2 for a plane layer, but find α 2/3 in a square subdomain D where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain D, the exponent measuring the decay of the scalar field is α 1/2 for a slip flow condition and α 3/4 for a no-slip condition. The scaling law exponents α for chaotic timeperiodic flows are compared with those for similarly constructed random flows.

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Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

Cited by 6 publications

References 47 publications

Analysis of passive scalar advection in parallel shear flows: Sorting of modes at intermediate time scales

Analysis of passive scalar advection in parallel shear flows: Sorting of modes at intermediate time scales

Averaged dynamics of time-periodic advection diffusion equations in the limit of small diffusivity

Passive scalar decay in chaotic flows with boundaries

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