The p-th order structure function S_p(ℓ) = ⟨|u(x+ℓ) − u(x)|^p⟩ measures velocity increment statistics at scale ℓ. Kolmogorov (K41) predicts ζ_p = p/3 (straight line), but intermittency causes anomalous exponents: ζ_p deviates from linearity for p > 3. The third-order structure function is exact: S_3(ℓ) = −(4/5)εℓ (Kolmogorov 4/5 law, exact in the inertial range). The She-Leveque model predicts ζ_p = p/9 + 2(1−(2/3)^(p/3)), matching experiments remarkably well.
| p | ζ_p (K41) | ζ_p (SL94) | ζ_p (LN) | ζ_p (measured) |
|---|
S_p(ℓ) on log-log — slopes = ζ_p
Scaling exponents ζ_p — K41 vs She-Leveque vs measured
Turbulent velocity signal used for structure function computation