Eddy diffusivities in scalar transport

TL;DR

This paper investigates standard and anomalous scalar transport in incompressible flows using multiscale analysis, deriving bounds for eddy-diffusivities, and comparing theoretical predictions with numerical simulations to identify anomalous diffusion.

Contribution

It introduces a multiscale approach to compute eddy-diffusivities, including bounds and perturbative solutions, and demonstrates how anomalous diffusion manifests at small molecular diffusivities.

Findings

Derived an upper bound for eddy-diffusivities applicable to static and time-dependent flows.

Used perturbative expansion and numerical methods to solve the auxiliary problem.

Identified signatures of anomalous diffusion through singular behavior of eddy-diffusivity.

Abstract

Standard and anomalous transport in incompressible flow is investigated using multiscale techniques. Eddy-diffusivities emerge from the multiscale analysis through the solution of an auxiliary equation. From the latter it is derived an upper bound to eddy-diffusivities, valid for both static and time-dependent flow. The auxiliary problem is solved by a perturbative expansion in powers of the P\'eclet number resummed by Pad\'e approximants and by a conjugate gradient method. The results are compared to numerical simulations of tracers dispersion for three flows having different properties of Lagrangian chaos. It is shown on a concrete example how the presence of anomalous diffusion can be revealed from the singular behaviour of the eddy-diffusivity at very small molecular diffusivities.