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Ruwim Berkowicz
and
Lars P. Prahm

Abstract

The gradient transfer theory for turbulent diffusion is reformulated in order to obtain an improved method for applied dispersion studies. The basic innovation is that diffusivity of single Fourier components of the concentration field is treated separately, i.e., spectral turbulent diffusivity coefficients are introduced. The value of the diffusivity decreases with increasing wave vector k of the concentration spectrum. The rate of growth of an expanding cloud of material thus becomes dependent on the stage of growth. This is in qualitative agreement with the statistical dispersion theory. It is shown that the assumption of k-dependent diffusivity leads to a nonlocal flux-gradient relation. A new function, the turbulent diffusivity transfer function, is introduced. The turbulent diffusive flux depends on concentration gradients at all points in the space. The diffusion equation is written in terms of the turbulent diffusivity transfer function. The width of the turbulent diffusivity transfer function is shown to determine the validity of the traditional gradient transfer theory formulation. The turbulent dispersion can be considered as Gaussian when the size of the cloud is considerably larger than the size of the most energetic turbulent eddies. These eddies determine the width of the turbulent diffusivity transfer function.

In general, it is shown that the shape of the cloud is non-Gaussian and the width, computed in terms of spectral turbulent diffusivity coefficients, is smaller than in a Gaussian distribution. This deviation decreases with increasing size of the cloud.

The present theory reveals properties in agreement with experiments and Lagrangian statistical dispersion theory and has the advantage of being an Eulerian method which can be used for air pollution dispersion models treating multiple, interacting sources.

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Lars P. Prahm
,
Ruwim Berkowicz
, and
Ove Christensen

Abstract

Further development of the spectral turbulent diffusivity concept is presented with the aim of obtaining an Eulerian dispersion model applicable for multiple interacting sources. The theory is applied for studies of plume dispersion in a field of a homogeneous and stationary turbulence. A continuous plume is considered as consisting of an infinite number of expanding puffs. The puffs' center of mass fluctuates following the long-wave range of the turbulent velocity fluctuation spectrum. The center-of-mass fluctuations are assigned to phases of the Fourier coefficients of the concentration distribution. The standard deviation of the velocity of the phase fluctuations is dependent on the wave vector of the Fourier coefficient. Time-averaging results in a spectral phase diffusivity coefficient.

It is shown that the rate of growth and the center-line concentration obtained by the spectral diffusivity model are in agreement with results predicted by the Lagrangian statistical theory. For a narrow plume, it is shown that the plume width is proportional to the time of travel, while for a narrow puff, the 3/2-power dependence is found. For a narrow distribution, the concentration shape deviates, however, from a Gaussian shape, in contradiction to results of the statistical theory.

It is shown that only two external parameters are required in the spectral turbulent diffusivity model. These are the long-wave range diffusivity coefficient K 0 and the wave vector k m of the most energetic turbulent eddies. An Eulerian integro-differential transport equation is the final result of the model. This equation can also be used for dispersion in case of space- and time-dependent parameters. We suggest a procedure for a direct experimental test of the spectral turbulent diffusivity concept.

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