Lagrangian structure functions in fully-developed hydrodynamical turbulence

TL;DR

This paper derives and analytically calculates the scaling exponents of Lagrangian velocity structure functions in fully developed turbulence, confirming their agreement with experimental data and introducing universal Lagrangian position structure functions.

Contribution

It provides a theoretical derivation of Lagrangian structure functions and introduces the concept of Lagrangian position structure functions with universal scaling.

Findings

Scaling exponents $\

are in agreement with experimental results.

Lagrangian position structure functions $R_n( au)$ have universal scaling for $n>3$.

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.