In its primitive form the Kolmogorov theory states that the four-fifths law can be generalized to higher-order structure functions according to relation (11.33) by assuming self-similarity. Experiments and numerical simulations clearly show a discrepancy from this prediction (see Figure 11.10): this is what is commonly called intermittency. Even if intermittency remains a still poorly understood property of turbulence because it still challenges any attempt at a rigorous analytical description from first principles (i.e. the Navier–Stokes equations), several models have been proposed to reproduce the statistical measurements, of which the simplest is probably the fractal model, also called the β model, which was introduced in 1978 (Frisch et al., 1978). As we shall see, this model is based on the idea of a fractal (incompressible) cascade and is therefore inherently a self-similar model. However, because the structure-function exponents are not those predicted by the Kolmogorov theory, one speaks of intermittency and anomalous exponents. Refined models have also been proposed, and we will present in this chapter the two most famous models: the log-normal and log-Poisson models.

Intermittency

Fractals and Multi-fractals

The idea underlying the β fractal model is Richardson's cascade (Figure 11.7): at each step of the cascade the number of children vortices is chosen so that the volume (or the surface in the two-dimensional case) occupied by these eddies decreases by a factor β (0 < β < 1) compared with the volume (or surface) of the parent vortex. The β factor is a parameter less than one of the model to reflect the fact that the filling factor varies according to the scale considered: the smallest eddies occupy less space than the largest.

We define by ln the discrete scales of our system: the fractal cascade is characterized by jumps from the scale ln to the scale ln+1.We show an example of a fractal cascade in Figure 13.1: at each step of the cascade the elementary scale is divided by two.