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The modified cuinulant expansion for two-dimensional isotropic turbulence

Published online by Cambridge University Press:  20 April 2006

Tomomasa Tatsumi  and
Shinichiro Yanase
Tomomasa Tatsumi
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan
Shinichiro Yanase
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan Engineering Mathematics, School of Engineering, Okayama University, Okayama 700, Japan.
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Abstract

The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 110 , September 1981 , pp. 475 - 496
Copyright
© 1981 Cambridge University Press

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