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Statistical mechanics of the Burgers model of turbulence

Published online by Cambridge University Press:  29 March 2006

Tomomasa Tatsumi  and
Shigeo Kida
Tomomasa Tatsumi
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto, Japan
Shigeo Kida
Affiliation:
Department of Physics, Faculty of Science, University of Kyoto, Kyoto, Japan
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Abstract

The velocity field of the Burgers one-dimensional model of turbulence at extremely large Reynolds numbers is expressed as a train of random triangular shock waves. For describing this field statistically the distributions of the intensity and the interval of the shock fronts are defined. The equations governing the distributions are derived taking into account the laws of motion of the shock fronts, and the self-preserving solutions are obtained. The number of shock fronts is found to decrease with time t as t−α, where α (0 [les ] α < 1) is the rate of collision, and consequently the mean interval increases as tα. The distribution of the intensity is shown to be the exponential distribution. The distribution of the interval varies with α, but it is proved that the maximum entropy is attained by the exponential distribution which corresponds to α = ½. For α = ½, the turbulent energy is shown to decay with time as t−1, in good agreement with the numerical result of Crow & Canavan (1970).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 55 , Issue 4 , 24 October 1972 , pp. 659 - 675
Copyright
© 1972 Cambridge University Press

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References

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