# Kolmogorov's second similarity hypothesis #turbulence #in-progress >[!quote] In the case of turbulence with sufficiently large Reynolds number $Re$, the multi-dimensional probability distributions for the relative velocities $\mathbf{v}(\mathbf{r}_k, \tau_k),\quad k=1,\dots,n$ in sufficiently small space and time intervals $|\mathbf{r}_k|\ll L$ and $|\tau_k|\ll L/U$ which satisfy the additional conditions >$ >\begin{align} >|\mathbf{r}_k|\gg \eta,\quad |\mathbf{r}_j - \mathbf{r}_k| \gg \eta \quad \textrm{when}\,\, j\neq k \\ >|\tau_k| \gg \tau_\eta,\quad |\tau_j - \tau_k| \gg \tau_\eta \quad \textrm{when}\,\, j\neq k >\end{align} >$ >are unambiguously determined by the value of $\overline{\varepsilon}$ and are independent of $\nu$. ## References 1. Monin, Yaglom - *Statistical Fluid Mechanics (Vol. 2)*