Lagrangian structure functions in fully-developed hydrodynamical turbulence

Abstract

The Lagrangian velocity structure functions in the inertial range of fully developed fluid turbulence are derived basing on the Navier-Stokes equations. For time $\tau$ much smaller than the correlation time, the structure functions are shown to obey the scaling relations $K_n(\tau)\propto \tau^{\zeta_n}$. The scaling exponents $\zeta_n$ are calculated analytically. The obtained values are in amazing agreement with the unique experimental results of the Bodenschatz group \cite{Bod2}. New notion -- the Lagrangian position structure functions $R_n(\tau)$ is introduced. All the $R_n$ of the order $n>3$ are shown to have a universal scaling.