Non-local dispersion and the reassessment of Richardson's t3-scaling law

G.E Elsinga,J.C.R Hunt,T Ishihara

doi:10.1017/jfm.2021.989

Abstract

The Richardson-scaling law states that the mean square separation of a fluid particle pair grows according to t3 within the inertial range and at intermediate times. The theories predicting this scaling regime assume that the pair separation is within the inertial range and that the dispersion is local, which means that only eddies at the scale of the separation contribute. These assumptions ignore the structural organization of the turbulent flow into large-scale shear layers, where the intense small-scale motions are bounded by the large-scale energetic motions. Therefore, the large scales contribute to the velocity difference across the small-scale structures. It is shown that, indeed, the pair dispersion inside these layers is highly non-local and approaches Taylor dispersion in a way that is fundamentally different from the Richardson-scaling law. Also, the layer's contribution to the overall mean square separation remains significant as the Reynolds number increases. This calls into question the validity of the theoretical assumptions. Moreover, a literature survey reveals that, so far, t3 scaling is not observed for initial separations within the inertial range. We propose that the intermediate pair dispersion regime is a transition region that connects the initial Batchelor- with the final Taylor-dispersion regime. Such a simple interpretation is shown to be consistent with observations and is able to explain why t3 scaling is found only for one specific initial separation outside the inertial range. Moreover, the model incorporates the observed non-local contribution to the dispersion, because it requires only small-time-scale properties and large-scale properties.

Highlights

The relative dispersion of two tracer particles in a turbulent flow has received considerable interest because it is a tractable problem closely connected to turbulent mixing
Either approximate t3 scaling was obtained for a single initial separation outside the inertial range (r0 < 60η) or observed only for very short times, much less than a decade
The common theories predicting this t3-scaling regime assume that only the eddies at the scale of the pair separation contribute to the dispersion

Summary

Introduction

The relative dispersion of two tracer particles in a turbulent flow has received considerable interest because it is a tractable problem closely connected to turbulent mixing. The spatial correlations, including the variance, of a passive scalar are related to the statistics of the relative pair separation (Batchelor 1952; Monin & Yaglom 1975). The separation distance, given by r(t) = |r(t)|, is widely believed to scale as r2 ∼ t3 in the inertial range and for intermediate times, which is referred to as Richardson scaling. Indicates averaging over a large ensemble of pairs. Richardson (1926) obtained the t3 scaling by proposing a diffusion equation for relative dispersion in isotropic turbulence, where, based on his experimental observations, the diffusion coefficient, K, was scale dependent according to K ∼ r4/3

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