Journal ArticleSelf-regularization in turbulence from the Kolmogorov 4/5-law and alignmentDrivas, Theodore D.A defining feature of three-dimensional hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon—anomalous dissipation—is sometimes called the ‘zeroth law of turbulence’ as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions S p ( ℓ ) := ⟨ | δ ℓ u | p ⟩ exhibit persistent power-law scaling in the inertial range, namely S p ( ℓ ) ∼ | ℓ | ζ p for exponents ζ p > 0 over an ever increasing (with Reynolds) range of scales. This behaviour indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that ζ p = p / 3 for all p and Onsager’s 1949 theory establishes the requirement that ζ p ≤ p / 3 for p ≥   3 for consistency with the zeroth law. Empirically, ζ 2 ⪆ 2 / 3 and ζ 3 ⪅ 1 , suggesting that turbulent Navier–Stokes solutions approximate dissipative weak solutions of the Euler equations possessing (nearly) the minimal degree of singularity required to sustain anomalous dissipation. In this note, we adopt an experimentally supported hypothesis on the anti-alignment of velocity increments with their separation vectors and demonstrate that the inertial dissipation provides a regularization mechanism via the Kolmogorov 4/5-law. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.2022-06-2710327734Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences38022261364-503Xhttps://doi.org/10.1098/rsta.2021.00332106233National Science Foundation