Decoupling of Eulerian and Lagrangia varables in Lagrangian velocity correlations

TL;DR

This paper derives an exact relationship between Lagrangian and Eulerian velocity correlations in fluid flow, showing how particle trajectories can be decoupled from the velocity field under certain statistical assumptions.

Contribution

It provides a theoretical framework that expresses Lagrangian velocity correlations in terms of Eulerian correlations and the distribution of particle displacements, under assumptions of randomness and stationarity.

Findings

Lagrangian velocity correlation can be exactly expressed via Eulerian correlations.

The decoupling is valid to leading order in system volume.

The approach applies to stationary, isotropic, translational invariant velocity fields.

Abstract

The motion of a particle carried by a liquid is described by the differential equation equating the velocity of the particle at time t to the the Eulerian velocity field at time t and at the location of the particle at that time. Assuming the velocity field to be random with a stationary, isotropic, translational invariant, zero mean distribution, the Lagrangian velocity correlation of the particle can be expressed in terms of the Eulerian correlations and the characteristic function of the probability distribution of the end point of the trajectory at time t, where the particle is taken to be at the origin at time 0. This is a result of a decoupling, which is exact to leading order in the volume of the system.