# Kolmogorov's hypotheses #turbulence #in-progress > [!quote] > **H1.** In the limit of infinite Reynolds numbers, all the possible symmetries of the Navier-Stokes equation, usually broken by the mechanisms producing the turbulent flow, are restored in a statistical sense at small scales and away from boundaries. > > **H2.** Under the same assumptions as in H1, the turbulent flow is self-similar at small scales, i.e. it possesses a unique scaling exponent $h$. $\exists h \in \mathbb{R} : \delta \mathbf{v}(\mathbf{r}, \lambda \mathbf{\ell}) = \lambda^h \delta\mathbf{v}(\mathbf{r}, \mathbf{\ell})$ > > **H3.** Under the same assumptions as in H1, the turbulent flow has a finite nonvanishing mean rate of dissipation $\epsilon$ per unit mass. ## References 1. Frisch, U. (1995). _Turbulence: the legacy of AN Kolmogorov_. Cambridge university press.