The Diffusion Approximation in Turbulent Two-Particle Dispersion

TL;DR

This paper develops a diffusion-based model to accurately reproduce the probability density functions of fluid particle pair separations in turbulence, linking it to Lagrangian velocity structure-functions and validating with numerical data.

Contribution

It introduces an inverse problem approach to derive a time-dependent diffusivity tensor from turbulence data, simplifying complex fluid dynamics into a diffusion equation.

Findings

Diffusion theory reproduces PDFs well at rms separations

Overpredicts growth rate of mean-square dispersion due to neglect of memory effects

Provides integral formulas for diffusivity based on turbulence hypotheses

Abstract

We solve an inverse problem for fluid particle pair-statistics: we show that a time sequence of probability density functions (PDF's) of separations can be exactly reproduced by solving the diffusion equation with a suitable time-dependent diffusivity. The diffusivity tensor is given by a time-integral of a conditional Lagrangian velocity structure-function, weighted by a ratio of PDF's. Physical hypotheses for hydrodynamic turbulence (sweeping, short memory, mean-field) yield simpler integral formulas, including one of Kraichnan and Lundgren. We evaluate the latter using a spacetime database from a numerical Navier-Stokes solution for driven turbulence. This diffusion theory reproduces PDF's well at rms separations, but growth rate of mean-square dispersion is overpredicted due to neglect of memory effects. More general applications of our approach are sketched.