A simple dynamical model of intermittent fully developed turbulence

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 ∼ l0 2−n only a fraction β n 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 〈δ v6(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.