dorsal/arxiv
View SchemaStirring up trouble: Multi-scale mixing measures for steady scalar sources
| Authors | Tiffany A. Shaw, Jean-Luc Thiffeault, Charles R. Doering |
|---|---|
| Categories | |
| ArXiv ID | physics/0607270 |
| URL | https://arxiv.org/abs/physics/0607270 |
| DOI | 10.1016/j.physd.2007.05.001 |
| Journal | Physica D 231, 143-164 (2007) |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
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.
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"abstract": "The mixing efficiency of a flow advecting a passive scalar sustained by\nsteady sources and sinks is naturally defined in terms of the suppression of\nbulk scalar variance in the presence of stirring, relative to the variance in\nthe absence of stirring. These variances can be weighted at various spatial\nscales, leading to a family of multi-scale mixing measures and efficiencies. We\nderive a priori estimates on these efficiencies from the advection--diffusion\npartial differential equation, focusing on a broad class of statistically\nhomogeneous and isotropic incompressible flows. The analysis produces bounds on\nthe mixing efficiencies in terms of the Peclet number, a measure the strength\nof the stirring relative to molecular diffusion. We show by example that the\nestimates are sharp for particular source, sink and flow combinations. In\ngeneral the high-Peclet number behavior of the bounds (scaling exponents as\nwell as prefactors) depends on the structure and smoothness properties of, and\nlength scales in, the scalar source and sink distribution. The fundamental\nmodel of the stirring of a monochromatic source/sink combination by the random\nsine flow is investigated in detail via direct numerical simulation and\nanalysis. The large-scale mixing efficiency follows the upper bound scaling\n(within a logarithm) at high Peclet number but the intermediate and small-scale\nefficiencies are qualitatively less than optimal. The Peclet number scaling\nexponents of the efficiencies observed in the simulations are deduced\ntheoretically from the asymptotic solution of an internal layer problem arising\nin a quasi-static model.",
"arxiv_id": "physics/0607270",
"authors": [
"Tiffany A. Shaw",
"Jean-Luc Thiffeault",
"Charles R. Doering"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1016/j.physd.2007.05.001",
"journal_ref": "Physica D 231, 143-164 (2007)",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Stirring up trouble: Multi-scale mixing measures for steady scalar sources",
"url": "https://arxiv.org/abs/physics/0607270"
},
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