An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows

Marco Avellaneda, Andrew J. Majda

Research output: Contribution to journalArticle

Abstract

Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Péclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Péclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Péclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves like k0(Pe)1/2 in the limit of large Péclet numbers where k0 is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

Original languageEnglish (US)
Pages (from-to)339-391
Number of pages53
JournalCommunications in Mathematical Physics
Volume138
Issue number2
DOIs
StatePublished - May 1991

Fingerprint

Laminar Flow
Diffusivity
Advection
advection
laminar flow
Turbulent Flow
Integral Representation
turbulent flow
diffusivity
Advection-diffusion
Velocity Field
Stieltjes integral
velocity distribution
Stieltjes Integral
perturbation
Peclet number
Perturbation Expansion
variational principles
Composite Materials
Variational Principle

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows. / Avellaneda, Marco; Majda, Andrew J.

In: Communications in Mathematical Physics, Vol. 138, No. 2, 05.1991, p. 339-391.

Research output: Contribution to journalArticle

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