Intermittency

The basic assumption of Kolmogorov K41 theory is the self-similarity of turbulent flow. Indeed experimental data [12,13] shows that the probability distribution function (pfd) of velocity increments are roughly Gaussian at large scales, but when small scales are considered, the tails of the distributions depart from the Gaussian behavior, and events larger than the standard deviation have a larger probability to occur than in the Gaussian case.

Figure 1.2: Probability distribution function $\sigma _{\ell } P(\delta u_{\ell })$ of longitudinal velocity increments at different scales $\ell$ normalized with their standard deviations $\sigma_{\ell} = \langle \delta u_{\ell}^2 \rangle^{1/2}$. Experiments by Herweijer and Van de Water [1995]

This anomaly is confirmed by time series of turbulent velocities, which appear to be roughly self similar, but once they are high-pass filtered reveals an intermittent behavior, i.e. there is an alternation of periods of quiescence and periods of intense fluctuations [5]. Within the Taylor's hypothesis this is equivalent to state that on small scales the statistic of velocity fluctuations is strongly intermittent.

A measure of the intermittency is given by the flatness:

$$F(\ell) = \frac {S_4(\ell)}{(S_2(\ell))^2} = {\langle (\delt... ...\ell})^4 \rangle \over \langle (\delta u_{\ell})^2 \rangle^2}$$ (1.55)

Since the fourth order moment receives contributions from the tails of the pdf, the flatness gives a measure of how frequent are the events larger than the standard deviation. While the flatness of turbulent velocity fluctuations on large scale is close to the Gaussian value $F=3$, it grows on small scale, as a consequence of the non-similarity of pdf's of fluctuations at different scales.

Measurements of the high order structure function[11] have shown that a power law behavior is indeed observable,

$$S_p(\ell) = \langle (\delta u_{\ell})^p \rangle \ell^{\zeta_p}$$ (1.56)

but the scaling exponents differ from the dimensional $K41$ prediction $\zeta_p = p/3$.

In the multifractal approach [14], instead of a global scale-invariance, a local scale-invariance is assumed, so that the scaling exponent $h$ of the velocity field can assume a whole range of different values, with a probability determined by the dimension $D(h)$ of the fractal set with a given exponent $h$. The scaling exponents of the structure function of order $p$ are obtained as Legendre transform of the fractal dimension $D(h)$:

$$\zeta_p = \inf_{h} [ph +3 - D(h)]$$ (1.57)