Total dispersion of a scalar quantity in turbulent flow

Abstract

A general approach to the problem of the dispersion of a scalar quantity in a turbulent flow is given. The essential feature of this formulation is the consideration of the motion of isoconcentration surfaces. Use of this continuum kinematic concept, while preserving the exact results for both the limiting cases of zero molecular diffusivity and no fluid motion, enables the study of the dispersion of scalar quantities, like heat, without the use of physical concepts which are inconsistent with the continuum field description. It is shown that the effect of the interaction of the molecular and turbulent diffusion processes is such that the total dispersion of the scalar is less than the sum of the individual dispersions due to fluid motion alone and the direct effect of molecular diffusivity. Of the two terms arising from the interaction, the first one obtained by a simple approximation to the position field is identical to that obtained by an entirely different method. The additional term which was not found in the earlier studies varies as the square of the time. A preliminary extension to the dispersion of a specie of negligible molecular diffusivity undergoing a first-order chemical reaction shows that this method can also be extended to other related problems.