Self-Consistent Field Theory for the Convective Turbulence in a Rayleigh-Benard System in the Infinite Prandtl Number Limit View Full Text


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Article Info

DATE

2015-06-28

AUTHORS

Jayanta K. Bhattacharjee

ABSTRACT

The kinetic energy spectrum Eu(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{u}(k)$$\end{document} for three dimensional convective turbulence in a Rayleigh-Benard system,where k is the wave vector, was shown to scale as k-13/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k ^{-13/3}$$\end{document} on heuristic grounds in the recent work of Pandey, Verma and Mishra in the infinite Prandtl number limit. They also presented clear numerical evidence of this scaling. This limit is very similar to the spherical model of critical phenomena and hence amenable to exact treatment in a self-consistent field theory. We find that self-consistency gives Eu(k)∝R22/15k-13/3(R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_u (k)\propto R^{22/15}k^{-13/3}(R$$\end{document} is the Rayleigh number) but the inevitable presence of sweeping adds a part which is proportional to k-7/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^{-7/2}$$\end{document}. This can account for the slight k-dependence of the compensated spectrum of Pandey et al. We also estimate the anisotropy in the spectrum and find that the second order Legendre function has a strength of 15 % relative to the isotropic part. In two spatial dimensions the scaling exponent of the energy spectrum is still 13/3 but the anisotropy is larger. More... »

PAGES

1519-1528

References to SciGraph publications

  • 2008. Turbulence in Fluids, Fourth Revised and Enlarged Edition in NONE
  • 2000-04. Turbulent convection at very high Rayleigh numbers in NATURE
  • 1986-03. Renormalization group analysis of turbulence. I. Basic theory in JOURNAL OF SCIENTIFIC COMPUTING
  • 1978-02. Planar diagrams in COMMUNICATIONS IN MATHEMATICAL PHYSICS
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    http://dx.doi.org/10.1007/s10955-015-1292-z

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