Real turbulence is a cascade: kinetic energy injected at the largest eddies passes down through a chain of shrinking eddies until it dissipates as heat at the tiny Kolmogorov scale. Across that inertial range the energy spectrum follows a strict power law, not a guess:
E(k) ∝ k⁻⁵³⁵ Kolmogorov -5/3 law
N ∼ Re⁵⁹⁵⁴ DNS cost (classic 3-D estimate)
νₛₛ₌ = (Cₛ Δ)² |S̄| Smagorinsky subgrid viscosity
k = ½ mean(u′²), νₜ = Cμ k²/ε k-ε eddy viscosity
This sim builds one real synthetic field from that spectrum: many plane-wave octaves at wavenumber k, random phase, amplitude ∝ k⁻⁵⁵ (so energy ∝ amplitude² ∝ k⁻⁵⁵⁵), each oriented perpendicular to its own wavevector so the sum stays divergence-free — the same random-Fourier-mode trick used in kinematic-simulation turbulence models.
- DNS shows every octave the current grid's Nyquist limit can represent — that is the entire cost of "resolve everything," and the required cell count to do it for a real flow grows as Re9/4, infeasible past a modest Re on any machine.
- LES keeps only octaves below a cutoff kc (a real spectral low-pass filter of width Δ = 1/kc) and models everything above it as an eddy viscosity computed from the filtered field's own strain rate.
- RANS throws away the instantaneous field entirely and keeps only its statistical mean over many realizations — steady, but with zero resolved turbulent structure; all of it is folded into k and νt.