k41

K41 refers to the Kolmogorov 1941 theory of turbulence, a foundational model in the statistical description of turbulent flows. Proposed by Andrey Kolmogorov in 1941, K41 seeks to describe homogeneous, isotropic turbulence at high Reynolds numbers by focusing on the inertial range—the range of scales between where energy is injected and where it is dissipated by viscosity.

The core ideas are that, in the inertial range, the statistics of small-scale fluctuations depend only on the mean energy dissipation rate per unit mass, ε, and on the wavenumber, k, but are independent of the specific large-scale forcing or fluid viscosity. This leads to the characteristic Kolmogorov spectrum: the energy density E(k) scales as E(k) = C_K ε^{2/3} k^{-5/3}, where C_K is a universal constant. Relatedly, the pth-order structure functions, S_p(r) = ⟨|u(x+r) − u(x)|^p⟩, are predicted to follow S_p(r) ∝ (ε r)^{p/3} in the inertial range, reflecting a self-similar energy cascade from large to small scales.

K41 rests on several simplifying assumptions, including fully developed, homogeneous, and isotropic turbulence, and a uniform energy flux in the inertial range. It provided a simple, widely used baseline for turbulence theory and modeling, influencing experimental analysis and computational approaches such as large-eddy and direct numerical simulations.

Limitations and refinements emerged over time. Empirical and theoretical work revealed intermittency effects that cause deviations from exact p/3 scaling, leading to Kolmogorov’s refined similarity hypotheses (K62) and subsequent models. Nonetheless, K41 remains a central reference point in turbulence research.