dorsal/arxiv
View SchemaMixing and coherent structures in two-dimensional viscous flows
| Authors | H. W. Capel, R. A. Pasmanter |
|---|---|
| Categories | |
| ArXiv ID | physics/0702040 |
| URL | https://arxiv.org/abs/physics/0702040 |
| DOI | 10.1016/j.physd.2008.04.016 |
| Journal | Physica D, 237 (2008), 1993-1997 |
Abstract
We introduce a dynamical description based on a probability density $\phi(\sigma,x,y,t)$ of the vorticity $\sigma$ in two-dimensional viscous flows such that the average vorticity evolves according to the Navier-Stokes equations. A time-dependent mixing index is defined and the class of probability densities that maximizes this index is studied. The time dependence of the Lagrange multipliers can be chosen in such a way that the masses $m(\sigma,t):=\intdxdy \phi(\sigma,x,y,t)$ associated with each vorticity value $\sigma$ are conserved. When the masses $m(\sigma,t)$ are conserved then 1) the mixing index satisfies an H-theorem and 2) the mixing index is the time-dependent analogue of the entropy employed in the statistical mechanical theory of inviscid 2D flows [Miller, Weichman & Cross, Phys. Rev. A \textbf{45} (1992); Robert & Sommeria, Phys. Rev. Lett. \textbf{69}, 2776 (1992)]. Within this framework we also show how to reconstruct the probability density of the quasi-stationary coherent structures from the experimentally determined vorticity-stream function relations and we provide a connection between this probability density and an appropriate initial distribution.
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"abstract": "We introduce a dynamical description based on a probability density\n$\\phi(\\sigma,x,y,t)$ of the vorticity $\\sigma$ in two-dimensional viscous flows\nsuch that the average vorticity evolves according to the Navier-Stokes\nequations. A time-dependent mixing index is defined and the class of\nprobability densities that maximizes this index is studied. The time dependence\nof the Lagrange multipliers can be chosen in such a way that the masses\n$m(\\sigma,t):=\\intdxdy \\phi(\\sigma,x,y,t)$ associated with each vorticity value\n$\\sigma$ are conserved. When the masses $m(\\sigma,t)$ are conserved then 1) the\nmixing index satisfies an H-theorem and 2) the mixing index is the\ntime-dependent analogue of the entropy employed in the statistical mechanical\ntheory of inviscid 2D flows [Miller, Weichman \u0026 Cross, Phys. Rev. A \\textbf{45}\n(1992); Robert \u0026 Sommeria, Phys. Rev. Lett. \\textbf{69}, 2776 (1992)]. Within\nthis framework we also show how to reconstruct the probability density of the\nquasi-stationary coherent structures from the experimentally determined\nvorticity-stream function relations and we provide a connection between this\nprobability density and an appropriate initial distribution.",
"arxiv_id": "physics/0702040",
"authors": [
"H. W. Capel",
"R. A. Pasmanter"
],
"categories": [
"physics.flu-dyn",
"cond-mat.other"
],
"doi": "10.1016/j.physd.2008.04.016",
"journal_ref": "Physica D, 237 (2008), 1993-1997",
"title": "Mixing and coherent structures in two-dimensional viscous flows",
"url": "https://arxiv.org/abs/physics/0702040"
},
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