Turbulence Intermittency & Multiscaling

Structure functions Sₚ(r) = ⟨|δu(r)|ᵖ⟩ ∝ r^ζₚ — anomalous exponents & departure from K41

Turbulence Model

μ (She-Lévêque):
ζ₂ (measured):
ζ₂ (K41=2/3): 0.667
ζ₆ (measured):
ζ₆ (K41=2): 2.000
Kolmogorov 1941 (K41): ζₚ = p/3
She-Lévêque (1994):
ζₚ = p/9 + 2[1−(2/3)^(p/3)]
Intermittency causes ζₚ to deviate
below K41 for p > 3.

Top: velocity increments δu(r)
Bottom: ζₚ vs p (K41, SL, measured)