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The multiple-scale cumulant expansion for isotropic turbulence

Published online by Cambridge University Press:  12 April 2006

Tomomasa Tatsumi ,
Shigeo Kida  and
Jiro Mizushima
Tomomasa Tatsumi
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan
Shigeo Kida
Affiliation:
Research Institute for Mathematical Sciences, University of Kyoto, Kyoto 606, Japan
Jiro Mizushima
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto 606, Japan Present address: Department of Mathematical Engineering, Sagami Institute of Technology, Fujisawa 251, Japan.
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Abstract

A method of multiple-scale expansion is applied to the theory of incompressible isotropic turbulence in order to close the infinite system of cumulant equations. The dynamical equation for the energy spectrum derived from this method is found to give positive-definite solutions at all Reynolds numbers. At large Reynolds numbers the spectrum takes the form of Kolmogorov's $-\frac{5}{3}$ power spectrum in the inertial subrange, whose extent increases indefinitely with Reynolds number. The spectrum in the energy-containing range satisfies an inviscid similarity law, so that the rate of energy decay or of viscous dissipation is also independent of the viscosity. In the higher wavenumber region beyond the inertial subrange the spectrum takes a universal form which is independent of its structure at lower wavenumbers. The universal spectrum is composed of three different subspectra, which are, in order of increasing wavenumber, the $k^{-\frac{5}{3}}$ spectrum, the k−1 spectrum and the exp [−σk1·5] spectrum, σ being a constant. Various statistical quantities such as the energy, the skewness of the velocity derivative, the microscale and the microscale Reynolds number are calculated from the numerical data for the energy spectrum. Theoretical results are discussed in detail in comparison with experimental results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 85 , Issue 1 , 7 March 1978 , pp. 97 - 142
Copyright
© 1978 Cambridge University Press

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