793 research outputs found
Stirring up trouble: Multi-scale mixing measures for steady scalar sources
The mixing efficiency of a flow advecting a passive scalar sustained by
steady sources and sinks is naturally defined in terms of the suppression of
bulk scalar variance in the presence of stirring, relative to the variance in
the absence of stirring. These variances can be weighted at various spatial
scales, leading to a family of multi-scale mixing measures and efficiencies. We
derive a priori estimates on these efficiencies from the advection--diffusion
partial differential equation, focusing on a broad class of statistically
homogeneous and isotropic incompressible flows. The analysis produces bounds on
the mixing efficiencies in terms of the Peclet number, a measure the strength
of the stirring relative to molecular diffusion. We show by example that the
estimates are sharp for particular source, sink and flow combinations. In
general the high-Peclet number behavior of the bounds (scaling exponents as
well as prefactors) depends on the structure and smoothness properties of, and
length scales in, the scalar source and sink distribution. The fundamental
model of the stirring of a monochromatic source/sink combination by the random
sine flow is investigated in detail via direct numerical simulation and
analysis. The large-scale mixing efficiency follows the upper bound scaling
(within a logarithm) at high Peclet number but the intermediate and small-scale
efficiencies are qualitatively less than optimal. The Peclet number scaling
exponents of the efficiencies observed in the simulations are deduced
theoretically from the asymptotic solution of an internal layer problem arising
in a quasi-static model.Comment: 37 pages, 7 figures. Latex with RevTeX4. Corrigendum to published
version added as appendix
Estimating eddy diffusivities from noisy Lagrangian observations
The problem of estimating the eddy diffusivity from Lagrangian observations
in the presence of measurement error is studied in this paper. We consider a
class of incompressible velocity fields for which is can be rigorously proved
that the small scale dynamics can be parameterised in terms of an eddy
diffusivity tensor. We show, by means of analysis and numerical experiments,
that subsampling of the data is necessary for the accurate estimation of the
eddy diffusivity. The optimal sampling rate depends on the detailed properties
of the velocity field. Furthermore, we show that averaging over the data only
marginally reduces the bias of the estimator due to the multiscale structure of
the problem, but that it does significantly reduce the effect of observation
error
Periodic Homogenization for Inertial Particles
We study the problem of homogenization for inertial particles moving in a
periodic velocity field, and subject to molecular diffusion. We show that,
under appropriate assumptions on the velocity field, the large scale, long time
behavior of the inertial particles is governed by an effective diffusion
equation for the position variable alone. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit . This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos
corrected. One reference adde
Resonant enhanced diffusion in time dependent flow
Explicit examples of scalar enhanced diffusion due to resonances between
different transport mechanisms are presented. Their signature is provided by
the sharp and narrow peaks observed in the effective diffusivity coefficients
and, in the absence of molecular diffusion, by anomalous transport. For the
time-dependent flow considered here, resonances arise between their
oscillations in time and either molecular diffusion or a mean flow. The
effective diffusivities are calculated using multiscale techniques.Comment: 18 latex pages, 11 figure
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