An Application of the Wiener-hermite Expansion to the Diffusion of a Passive Scalar in a Homogeneous Turbulent Flow

This re~Pa rch is supported hy thP Ad,·ancPd RPsParch Projects Agency under Contract :\o. DAHClS 67 C 011-l. Views or concluE<ions contained in this study should not be interpreted as representing the official opinion or policy of ARPA. · DISTRIBUTION ST ATEJ\IENT This document has been approved for public release and sale; its distribution is unlimited.This Rand Memorandum is presented as a competent treatment of the subject, worthy of publication. The Rand Corporation Youches for the quality of the research, without necessarily endorsing the opinions and conclusions of the author.PREFACE A difficult and recurrent problem in many applications of fluid mechanics is the description of the convection of a passive scalar quantity in a turbulent flow field. Examples of the diverse areas in which this problem arises are in reentry physics, when electron concentration is effected by wake turbulence, or in pollution studies, when the passive scalar, pollutant concentration, is transported by the turbulent flow field of the surrounding medium. A theoretical investigation of this problem is presented in this Memorandum, using a systematic approximation based on an expansion in Weiner-Hermite functionals. This approximation leads to analytic solutions to two crucial problems in the theory of turbulent diffusion: the diffusion of a passive scalar from a point source, and the spectrum of a statistically homogeneous scalar-both in a homogeneous turbulent flow.SUMMARY The velocity and concentration fields in a homogeneous turbulent flow convecting a passive scalar are expanded in series of Wiener-Hermite functionals. The equations of motion lead to integra-differential equations for the coefficients of the expansion. The expansion and the equations are truncated at the first stage that gives nontriv-ial results, and analytical solutions are obtained for the concentration field with the velocity field given and with zero molecular conductivity. The diffusion from a point source is considered first. It is shown that the mean concentration satisfies an integra-differential equation with diffusive and wave-like properties, which reduces for large time-values to the diffusion equation in accordance with the classical theory of G. I. Taylor. The effective diffusivity is determined in terms of Eulerian velocity correlations. The spectrum of a statistically homogeneous distribution of the scalar is considered second. It is shown that if the spectrum is initially concentrated around the wave number of the energy-containing eddies, an equilibrium is set up for larger wave numbers, with f(k) 2 ~ E(k) Ju where f(k) and E(k) are the concentration and energy …