Kolmogorov Turbulence — Scale Hierarchy

η = (ν³/ε)^(1/4) — from integral scale L to Kolmogorov microscale η, inertial range E(k) ~ k^(-5/3)

EDDY CASCADE SIMULATION

ENERGY SPECTRUM E(k)

Reynolds number Re 1000

Viscosity ν (×10⁻³) 1.0

Dissipation ε 1.0

L (integral scale): –

η (Kolmogorov): –

L/η (range): –

τ_η (Kolmogorov time): –

Re^(3/4) = L/η: –

Kolmogorov (K41) theory: In fully developed turbulence, energy is injected at large scales L (integral scale, Re-dependent), cascades through the inertial range without dissipation, and dissipates at the Kolmogorov microscale η = (ν³/ε)^(1/4) where viscous forces balance inertial forces. Key results: energy spectrum E(k) = C_K ε^(2/3) k^(-5/3) (Kolmogorov 1941); the ratio L/η ~ Re^(3/4); Kolmogorov time τ_η = (ν/ε)^(1/2); velocity at η is u_η = (νε)^(1/4). The inertial range spans wavenumbers 1/L to 1/η. The unique exact result: the 4/5 law ⟨(δu_∥)³⟩ = -4/5 εℓ (no free parameters). K62 corrections add intermittency via multifractal energy dissipation.