Telegraph-type versus diffusion-type models of turbulent relative dispersion

TL;DR

This paper compares two models of turbulent relative dispersion, the Richardson diffusion and the telegraph equation, analyzing their solutions and implications for particle separation in turbulence.

Contribution

It provides a detailed comparison of the solutions and properties of the Richardson diffusion and telegraph models, highlighting differences in their behavior and scaling laws.

Findings

The telegraph model predicts persistent particle separation.

The Richardson model's solution extends infinitely after initial time.

Batchelor scaling applies only to the telegraph model.

Abstract

Properties of two equations describing the evolution of the probability density function (PDF) of the relative dispersion in turbulent flow are compared by investigating their solutions: the Richardson diffusion equation with the drift term and the self-similar telegraph equation derived by Ogasawara and Toh [J. Phys. Soc. Jpn. 75, 083401 (2006)]. The solution of the self-similar telegraph equation vanishes at a finite point, which represents persistent separation of a particle pair, while that of the Richardson equation extends infinitely just after the initial time. Each equation has a similarity solution, which is found to be an asymptotic solution of the initial value problem. The time lag has a dominant effect on the relaxation process into the similarity solution. The approaching time to the similarity solution can be reduced by advancing the time of the similarity solution appropriately. Batchelor scaling, a scaling law relevant to initial separation, is observed only for the telegraph case. For both models, we estimate the Richardson constant, based on their similarity solutions.