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A simple dynamical model of intermittent fully developed turbulence

Published online by Cambridge University Press:  12 April 2006

Uriel Frisch ,
Pierre-Louis Sulem  and
Mark Nelkin
Uriel Frisch
Affiliation:
CNRS, Observatoire de Nice, France
Pierre-Louis Sulem
Affiliation:
CNRS, Observatoire de Nice, France
Mark Nelkin
Affiliation:
School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853
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Abstract

We present a phenomenological model of intermittency called the β-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales ∼ l02−n only a fraction βn of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974), is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turn-over times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation.

The β-model leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot's (1974, 1976) ‘fractal dimension’. For three-dimensional turbulence, the correction B to the $\frac{5}{3}$ exponent of the energy spectrum is equal to $\frac{1}{3}(3 - D)$ and is related to the exponent μ of the dissipation correlation function by $B = \frac{1}{3}\mu $ (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality $B \leqslant \frac{1}{3}\mu $. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second- and fourth-order moments have the same viscous cut-off.

The predictions of the β-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function 〈δv6(l)〉 and the dissipation correlation function 〈ε(r)ε(r + 1)〉 are related by \[ \langle \delta v^6(l)\rangle /l^2\sim\langle \epsilon({\bf r})\epsilon({\bf r}+{\bf l})\rangle \] in both models. We conjecture that this relation is model independent.

Finally, some possible directions for further numerical and experimental work on intermittency are indicated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 87 , Issue 4 , 29 August 1978 , pp. 719 - 736
Copyright
© 1978 Cambridge University Press

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