Turbulence and Kolmogorov 1941 (a heuristic theory)

This lesson is different from the others: most of what it states is heuristic, supported by experiments and computation rather than proved from the equations. At high Reynolds number flows become turbulent: irregular, unpredictable in detail, and full of eddies of every size. Richardson (1922) described the picture in verse: big whirls have little whirls that feed on their velocity. Energy is injected at a large scale \(L\), passed by the nonlinearity to smaller and smaller eddies, and dissipated by viscosity only at the smallest scales. The rate \(\varepsilon\) at which energy per unit mass flows down this cascade equals the rate at which it is dissipated.

Kolmogorov (1941, now called K41) made this quantitative with two hypotheses about statistically homogeneous, isotropic turbulence: at scales much smaller than \(L\) the statistics are universal and depend only on \(\varepsilon\) and \(\nu\); and at scales between \(L\) and the dissipation scale they depend on \(\varepsilon\) alone. Everything else is dimensional analysis. From \(\nu\) (m²/s) and \(\varepsilon\) (m²/s³) there is exactly one length, velocity and time: \[ \eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}, \qquad u_\eta = (\nu\varepsilon)^{1/4}, \qquad \tau_\eta = \left(\frac{\nu}{\varepsilon}\right)^{1/2}. \] The Kolmogorov length \(\eta\) is where eddies are small enough for viscosity to destroy them; its Reynolds number \(u_\eta\eta/\nu\) is exactly 1. With \(\nu = 10^{-6}\) m²/s and \(\varepsilon = 10^{-3}\) W/kg, \(\eta \approx 0.18\) mm.

In the inertial range, the energy spectrum \(E(k)\) (energy per unit wavenumber, units m³/s²) can only be built from \(\varepsilon\) and \(k\), which forces \[ E(k) = C\,\varepsilon^{2/3}k^{-5/3}, \] the famous five-thirds law, confirmed in tidal channels, wind tunnels and the atmosphere with \(C \approx 1.5\). Equivalently velocity differences over a distance \(r\) scale like \((\varepsilon r)^{1/3}\). Using \(\varepsilon \sim U^3/L\), the ratio of largest to smallest scale is \(L/\eta \sim \mathrm{Re}^{3/4}\), so a direct simulation resolving every eddy needs about \(\mathrm{Re}^{9/4}\) grid points: a reason turbulence remains computationally hard.

Two results stand out as more than dimensional analysis. Kolmogorov's four-fifths law, \(\langle(\delta u_\parallel)^3\rangle = -\tfrac45\varepsilon r\), is derived from the Navier-Stokes equations under assumptions of homogeneity, isotropy and a non-vanishing dissipation limit. And Onsager's conjecture (1949) that energy is conserved by Euler solutions with Hölder regularity above \(1/3\), but can be dissipated below it, is now a theorem: the conservation half by Constantin, E and Titi (1994) and the dissipation half by Isett (2018) and Buckmaster, De Lellis, Székelyhidi and Vicol (2019). The exponent \(1/3\) is exactly the K41 scaling of velocity increments. The K41 predictions themselves are not theorems, and measured higher-order statistics deviate from them (intermittency, addressed by Kolmogorov in 1962).

Picture it: a waterfall of energy tumbling from big eddies to small ones at a steady rate, with viscosity waiting at the bottom. Think it: dimensional analysis with a single conserved flux fixes every exponent; the mathematical content of turbulence theory is to find out which of these predictions actually follow from the equations.