Kolmogorov's K41 theory predicts structure function scaling ζ_p = p/3. But real turbulence shows anomalous scaling: the energy dissipation is not uniformly distributed but concentrated in intermittent bursts on fractal sets. The multifractal model (Parisi-Frisch 1985) describes this via a spectrum of Hölder exponents D(h), where h is the local velocity scaling. The She-Leveque (1994) formula ζ_p = p/9 + 2(1 − (2/3)p/3) gives excellent agreement with experiments.
Scaling exponents ζ_p vs order p
Multifractal spectrum D(h) — Legendre transform
Intermittent turbulent signal (bursts of activity)