A Generalized Brownian Motion Model for Turbulent Relative Particle Dispersion

TL;DR

This paper introduces a generalized Brownian motion model for turbulent particle dispersion, linking pressure force fluctuations to flow Reynolds number and reconciling various turbulence theories and scaling laws.

Contribution

It proposes a novel R_e-dependent model connecting pressure forces, energy dissipation, and particle dispersion, extending existing turbulence theories with explicit relations.

Findings

Derives an R_e-dependent Richardson-Obukhov constant g.

Shows agreement with Batchelor-Townsend scaling at low R_e.

Provides a unified framework connecting pressure fluctuations and turbulence parameters.

Abstract

In this paper, a generalized Brownian motion model has been applied to describe the relative particle dispersion problem in more realistic turbulent flows. The fluctuating pressure forces acting on a fluid particle are taken to be a colored noise and follow a stationary process and are described by the Uhlenbeck-Ornstein model while it appears plausible to take their correlation time to have a power-law dependence on the flow Reynolds number $R_e$, thus introducing a bridge between the Lagrangian quantities and the Eulerian parameters for this problem. This ansatz is in qualitative agreement with the possibility of a connection speculated earlier by Corrsin [26] between the white-noise representation for the fluctuating pressure forces and the large-$R_e$ assumption in the Kolmogorov [4] theory for the 3D fully developed turbulence (FDT) as well as the argument of Monin and Yaglom [23] and the result of Sawford [13] and Borgas and Sawford [24] that the Lagrangian acceleration is delta-function auto-correlated in the infinite-$R_e$ limit. It also provides an insight into the result that the Richardson-Obukhov scaling holds only in the infinite-$R_e$ limit and disappears otherwise. This ansatz further confirms the Lin-Reid [18] conjecture regarding the connection between the fluctuating pressure-force parameter and the energy dissipation rate in turbulence and leads to an $R_e$-dependent explicit relation between the two speculated by Lin and Reid [18]. More specifically, this ansatz provides a determination of the Richardson-Obukhov constant $g$ as a function of $R_e$, with an asymptotic constant value in the infinite-$R_e$ limit. It is shown to lead to full agreement, in the small-$R_e$ limit as well, with the Batchelor-Townsend [27] scaling for the rate of change of the mean square interparticle separation in 3D FDT, hence validating its soundness further.