A CORRECTED MIXING-LENGTH THEORY OF TURBULENT DIFFUSION

Alison M. Grant University of Melbourne

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Abstract

The results of the conventional mixing-length theory disagree with those of the statistical theory of turbulence, and must be rejected also on common-sense grounds since they imply the existence of infinite velocities. This does not result from any lack of realism on the part of the basic assumptions of the mixing-length theory, but from an inconsistency in their application, namely the implicit assumption that fluid parcels can travel the entire distance between “collisions” during an infinitesimal dispersion time, although they are explicitly assumed to travel with a finite velocity. Removal of this inconsistency leads to a corrected theory whose results are in agreement with those of the statistical theory. The corrected theory, which yields a diffusion equation with the coefficient K varying with dispersion time, also provides a description of the diffusion for all dispersion times in terms of the energy spectrum of eddy lifetimes. The criticism of the conventional mixing-length theory is shown to apply also to the classical theory of molecular diffusion, so that there can be no theoretical justification for the simple diffusion equation with K independent of the dispersion time. The implications of the corrected theory for atmospheric diffusion are discussed qualitatively, the diffusion of smoke from a source at ground level being taken as an example.

Abstract

The results of the conventional mixing-length theory disagree with those of the statistical theory of turbulence, and must be rejected also on common-sense grounds since they imply the existence of infinite velocities. This does not result from any lack of realism on the part of the basic assumptions of the mixing-length theory, but from an inconsistency in their application, namely the implicit assumption that fluid parcels can travel the entire distance between “collisions” during an infinitesimal dispersion time, although they are explicitly assumed to travel with a finite velocity. Removal of this inconsistency leads to a corrected theory whose results are in agreement with those of the statistical theory. The corrected theory, which yields a diffusion equation with the coefficient K varying with dispersion time, also provides a description of the diffusion for all dispersion times in terms of the energy spectrum of eddy lifetimes. The criticism of the conventional mixing-length theory is shown to apply also to the classical theory of molecular diffusion, so that there can be no theoretical justification for the simple diffusion equation with K independent of the dispersion time. The implications of the corrected theory for atmospheric diffusion are discussed qualitatively, the diffusion of smoke from a source at ground level being taken as an example.

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