Lagrangian statistics of particle pairs in homogeneous isotropic turbulence

Abstract

We present a detailed investigation of the particle pair separation process in homogeneous isotropic turbulence. We use data from direct numerical simulations up to R ϳ 280 following the evolution of about two million passive tracers advected by the flow over a time span of about three decades. We present data for both the separation distance and the relative velocity statistics. Statistics are measured along the particle pair trajectories both as a function of time and as a function of their separation, i.e., at fixed scales. We compare and contrast both sets of statistics in order to gain insight into the mechanisms governing the separation process. We find very high levels of intermittency in the early stages, that is, for travel times up to order ten Kolmogorov time scales. The fixed scale statistics allow us to quantify anomalous corrections to Richardson diffusion in the inertial range of scales for those pairs that separate rapidly. It also allows a quantitative analysis of intermittency corrections for the relative velocity statistics.

Self-similar two-particle separation model

“…The stretching rate has been measured, both in direct numerical simulations and with particle tracking velocimetry. 14,21,51,[53][54][55] It has been found that the nondimensionalized stretching rate…”

“…Kinematic simulations ͑KS͒ of turbulent-like flows [7][8][9] offers the possibility to study a number of dispersion aspects, also at high Reynolds numbers. Only recently [10][11][12][13][14][15] has it become possible to study the second moment of the probability density function ͑PDF͒ of the separation r͑t͒ between two particles also in real turbulence via direct numerical simulation ͑DNS͒ or experiment, with the notable exception of Bourgoin et al 15 at typically low to intermediate Reynolds numbers. Relative dispersion of fluid particles still serves as a benchmark problem for our current understanding of turbulence.…”

“…A number of experiments support the existence of the two regimes. [11][12][13][14][15]21 To date, the precise value of the Richardson constant g is still not known. Experimental values for g are difficult to obtain and have a considerable range.…”

“…11,21 For DNS simulations at Re = 200 − 283, values of g Ϸ 0.5− 0.7 are reported. [12][13][14]21 The influence on dispersion dynamics of finite Re flows is still subject of discussion. Exit time statistics introduced by Artale et al 37 measure separation velocities at fixed scales.…”

“…For a thorough theoretical overview on particle diffusion, we refer to Falkovich et al 20 Different assumptions on the diffusion coefficient were made and experiments were conducted in order to evaluate the proposed choices. Recently, particle tracking 11,21 and numerical 14 experiments showed that the shape of the separation PDF is approximated best by the Richardson model. This finding was again questioned, though, in Ref.…”