Extended universality in moderate-Reynolds-number flows

In the inertial interval of turbulence one asserts that the velocity structure functions ${\mathit{S}}_{\mathit{n}}$(r) scale like ${\mathit{r}}_{\mathit{n}}^{\mathit{n}\mathrm{\zeta }}$. Recent experiments indicate that ${\mathit{S}}_{\mathit{n}}$(r) has a more general universal form [rf(r/η)${]}_{\mathit{n}}^{\mathit{n}\mathrm{\zeta }}$, where η is the Kolmogorov viscous scale. This form seems to be obeyed on a range of scales that is larger than power law scaling. It is shown here that this extended universality stems from the structure of the Navier-Stokes equations and from the property of the locality of interactions. The approach discussed here allows us to estimate the range of validity of the universal form. In addition, we examine the possibility that the observed deviations from the classical values of ${\mathrm{\zeta }}_{\mathit{n}}$=1/3 are due to the finite values of the Reynolds numbers and the anisotropy of the excitation of turbulence.