Kolmogorov Turbulence — K41 Energy Cascade

Kolmogorov's 1941 theory predicts that in the inertial range (L ≫ ℓ ≫ η), energy cascades from large to small scales with spectrum E(k) = C_K ε^(2/3) k^(−5/3), where ε is the energy dissipation rate. The unique exact result from the Navier-Stokes equations is the 4/5 law: ⟨(δu∥)³⟩ = −4εr/5. Intermittency corrections (K62) give anomalous scaling ζ_p ≠ p/3, visible in the deviation of structure function exponents from the K41 prediction.

Kolmogorov η
Reynolds Re = (L/η)^(4/3)
ζ_p (K62) vs p/3
Inertial range decades